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A Fourier mode exits the Hubble horizon during inflation when . On superhorizon scales , the growing mode of a sufficiently light scalar perturbation approaches a constant.
A pencil generated by cubic forms is the one-parameter family indexed by . A general pair has nine transverse base points.
A chiral transformation rotates a Dirac field by . A massless Dirac kinetic term is invariant under a global rotation, while a mass term is not.
The strong-coupling scale of an asymptotically free gauge theory is the renormalization-group invariant scale at which the one-loop running coupling formally diverges. It replaces a dimensionless ultraviolet coupling through dimensional transmutation.
The Hilbert basis is the unique minimal generating set of the additive monoid of lattice points in a pointed rational cone.
A Munk boundary layer closes a Sverdrup balance interior using lateral viscosity. At a zonal wall, balancing with gives the width .
Weak existence and uniqueness in law for an additive-noise SDE with bounded drift by 
Codex 0 2026-09-24
Codex 0 2026-09-24If is bounded and measurable, then on every finite time intervalhas a weak solution and uniqueness in law. Starting with Wiener measure, the Novikov condition and Girsanov theorem add the drift. Applying the inverse change of measure to any weak solution recovers Wiener measure and identifies its law by the same pathwise density.
The empirical distribution function converges uniformly almost surely to the population distribution function.
A gravitational wave in four-dimensional general relativity has two independent linear polarizations, conventionally called plus and cross. A rotation by angle in the transverse plane rotates their pair by .
A symplectic sphere of self-intersection can be rationally blown down by taking the symplectic sum with along the sphere and a smooth conic of self-intersection . Equivalently, its disk-bundle neighborhood is replaced by the rational ball whose boundary is the lens space .
If a symplectomorphism between compact symplectic submanifolds lifts to an isomorphism of their symplectic normal bundles, then it extends to a symplectomorphism between neighborhoods. Thus the germ of a symplectic neighborhood is determined by the restricted form and the symplectic normal bundle.
When depth varies but the Coriolis parameter is constant, forcing can balance advection of rather than advection of planetary vorticity. This topographic potential-vorticity gradient plays the role normally played by the beta effect.
For a finite Markov chain with stationary distribution , the separation distance from an initial state isIf is a strong stationary time, then , and total variation distance is at most separation distance.
A pure state is an extreme point of the state space. The Krein-Milman theorem guarantees pure states, and every positive element attains its norm at some pure state.
If a smooth maximal cone is generated by a lattice basis , its star subdivision through is the fan of the blowup at the corresponding torus-fixed point.
If , then is an abelian subgroup of the unit group. Multiplication satisfies , and , so identifies the additive group of with .
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