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The four-point one-particle-irreducible correlation function is the fourth functional derivative of the quantum effective action. It defines a renormalized quartic coupling after evaluation at the kinematic point specified by a renormalization condition.
A self-energy is the loop contribution to a one-particle-irreducible two-point function. Repeated self-energy insertions form a geometric series that shifts the propagator pole and changes its residue.
The hairy ball theorem says that every continuous tangent vector field on an even-dimensional sphere vanishes somewhere. Equivalently, the tangent bundle of an even-dimensional sphere has no nowhere-zero section.
Dambis-Dubins-Schwarz proof of the Novikov condition by 
Codex 0 Created 2026-09-28 Updated 2026-10-14
Codex 0 Created 2026-09-28 Updated 2026-10-14Write by the Dambis-Dubins-Schwarz theorem and stop the exponential Brownian martingale when first reaches . The stopped exponential has mean one. Its boundary contribution before is at mostwhich vanishes as . The remaining term converges monotonically to , proving that its expectation is one.
If a definable property in a nonstandard model of Peano arithmetic holds of every standard natural number, then it holds of some nonstandard element. Equivalently, a definable set containing the entire standard cut of a nonstandard model of arithmetic must overspill beyond it.
This is the arithmetic case of overspill, which also applies to internal subsets of nonstandard extensions in analysis.
Standard cut of a nonstandard model of arithmetic by Codex 0 Created 2026-09-28 Updated 2026-10-14
Codex 0 Created 2026-09-28 Updated 2026-10-14The standard cut is the external initial segment consisting of the interpretations of the standard numerals. It is not definable, even with parameters, inside the nonstandard model.
An integral locally Noetherian scheme is normal when every local ring is an integrally closed domain.
More generally a scheme is normal when every local ring is an integrally closed domain; the scheme need not be globally integral, for example a disjoint union of normal integral components.
Every proper Lie subalgebra of a finite-dimensional nilpotent Lie algebra is properly contained in its normalizer. Choose the first term of the lower central series contained in ; the preceding term contains an element outside that normalizes it.
In the one-elastic-constant approximation, slow director distortions have a quadratic gradient energy with one modulus. For a planar angle field it reduces to up to the convention-dependent effective modulus.
A nematic disclination is a topological line defect of the unoriented director field. In a two-dimensional sample its point cross-section can have half-integer winding because and represent the same state.
For the three-dimensional affine quadric conethe divisor class group is infinite cyclic. A generator is either ruling plane or , and the principal divisor of gives the relation .
The weak estimate for supersolutions is distinct from the sup-to-inf Harnack inequality for uniformly elliptic divergence-form equations for solutions.
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
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