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A gauge coupling is the constant multiplying the gauge potential in the gauge covariant derivative. Its normalization depends on that of the Lie-algebra generators; each simple or Abelian factor of a gauge algebra may have an independent coupling.
The gauge group is the group of local internal symmetries. Its Lie algebra labels the gauge-boson species, while a matter field carries a representation of the group.
A gauge boson is a spin-one quantum excitation of a gauge field. Each generator of the gauge group supplies one gauge-boson species before symmetry breaking.
For irreducible , the Galois group is exactly when is a square in . In the transitive Klein-four action on the roots , all three nonidentity double transpositions fix , forcing it into the fixed field .
If a separable cubic has polynomial discriminant , then adjoining gives the subfield fixed by the alternating part of its Galois group. When the cubic Galois group is , this is its unique quadratic subfield.
A partial function from to assigns at most one value in to each element of ; it may be undefined outside a subset of .
When the energy line lies below the maximum of the pressureless-matter effective potential, its large- allowed region is separated from the Big-Bang region. A contracting large universe reaches the outer turning point at positive scale factor and re-expands, producing a nonsingular bounce.
For positive spatial curvature and positive cosmological constant, the effective potential has its maximum atThe critical case hasand contains the Einstein static universe.
Oscillatory Gaussian integral with negative phase by 
Codex 0 Created 2026-10-03 Updated 2026-10-07
Codex 0 Created 2026-10-03 Updated 2026-10-07A frailty random variable is the positive latent multiplier applied to an individual's hazard. Its scale is conventionally normalized, often by setting , because it is confounded with the scale of the baseline hazard.
Fourier coefficient decay of an analytic periodic function by Codex 0 Created 2026-10-03 Updated 2026-10-06
Codex 0 Created 2026-10-03 Updated 2026-10-06If a periodic function extends analytically and remains bounded in a nonzero complex strip around the real axis, contour shifting gives constants such thatThis exponential decay underlies spectral convergence of Fourier approximation and periodic trapezoidal quadrature.
The Fermi interaction is the low-energy four-fermion approximation to the weak charged current. Integrating out a W boson gives its coefficient at tree level.
The finite-condition Lévy collapse consists of finite partial functions with and , ordered by reverse inclusion. The generic union gives a surjection for every infinite . If is regular and uncountable, the Delta-system lemma at a regular uncountable cardinal proves the -chain condition, so the extension makes .
The punctured circular coneis geodesically incomplete because a generator reaches the missing vertex in finite time. It is an inextendible embedded surface: any proper connected embedded extension would have to add the vertex, but its tangent planes have different limits as varies. Nevertheless its Gaussian curvature of a cone away from its vertex is identically zero.
For a quantum harmonic oscillator perturbed by , write . Sincefirst-order nondegenerate perturbation theory gives
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