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Let an abelian group carry a nonnegative quadratic height with finite bounded subsets. If is finite for some , repeatedly choosing representatives modulo reduces the height by a fixed factor until a bounded set is reached. The representatives and that finite bounded set generate .
For an elliptic curve over a number field and an integer , the quotient is finite. Combining this finite quotient with a height descent proves finite generation of .
T-duality identifies a compact boson of radius with one of reciprocal radius by exchanging momentum and winding numbers. Factors of restore the dimensional relation .
For a morphism of schemes , the sheaf of relative Kähler differentials is obtained by sheafifying the modules on affine charts .
Transitivity exact sequence for Kähler differentials by 
Codex 0 Created 2026-09-24 Updated 2026-10-14
Codex 0 Created 2026-09-24 Updated 2026-10-14On a finite undirected graph, the reversal of a loop-erased random walk from to has the law of a loop-erased random walk from to . The uniform-spanning-tree path representation proves the identity.
The diagonal morphism sends a point to the pair consisting of that point twice. Every diagonal of schemes is a locally closed immersion, and a morphism is separated exactly when its diagonal is a closed immersion.
For a linear smoother , its effective degrees of freedom are . This measures sensitivity of fitted values to responses even when is not an orthogonal projection.
One-nearest-neighbour classification assigns a query the label of its single nearest training observation. Under local regularity, its limiting conditional error is .
A growing hull is parameterized by half-plane capacity when . The factor two makes its Chordal Loewner equation take the conventional form .
For with ,Indeed, the imaginary part at the Brownian exit point is at most one and the probability of reaching the radius- neighbourhood containing the rectangle from is . Consequently has capacity although its diameter tends to two.
There is a universal constant such that every compact H-hull satisfiesTranslate and scale so the hull lies in a unit half-disc, then use the Brownian representation of half-plane capacity and the harmonic measure of that half-disc as viewed from .
If is the first exit time of planar Brownian motion from , thenIt follows by applying the optional sampling theorem for a supermartingale to the harmonic function .
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