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A Skyrmion is a finite-energy topological soliton of the Skyrme model. Its baryon number is the degree of the compactified spatial map .
Simon's algorithm prepares hidden-subgroup coset states, applies the quantum Fourier transform over , and samples vectors satisfying over . Repeated samples and Gaussian elimination recover with oracle queries and polynomial computation.
Every automorphism of a full matrix algebra is inner by 
Codex 0 Created 2026-09-28 Updated 2026-10-05
Codex 0 Created 2026-09-28 Updated 2026-10-05Every unital algebra automorphism has the form for some invertible . One proof transports the matrix units through , chooses compatible bases for their rank-one images, and reconstructs the common similarity transformation.
If , ordering the least prime factor giveswhere in a set formulation, or equivalently restricts the original sequence to terms divisible by .
Put . A nonnegative sequence has distribution when is multiplicative on the squarefree divisors of , , andfor every .
A Keplerian shearing sheet has background velocity , radial shear rate , and radial epicyclic frequency .
The stratified incompressible shearing sheet adds a buoyancy displacement with and acceleration . Axisymmetric plane waves obeyinterpolating between inertial and buoyancy oscillations.
A Kida vortex is an exact elliptical patch of uniform vorticity embedded in a linear shear flow. In a Keplerian shearing sheet its aspect ratio controls both the circulation period and the stability of three-dimensional perturbations.
The global section functor sends a sheaf on to . It is left exact, and its right derived functors are sheaf cohomology.
For a two-dimensional Newtonian ice sheet of thickness on a flat no-slip bed, hydrostatic pressure and vertical shear give the depth-integrated fluxwhere is the kinematic viscosity. Mass conservation is for accumulation rate .
In the Solovay rank-comparison game, Players I and II produce ; ill-founded codes lose before ranks are compared, and among well-order codes the prescribed inequality between and decides the winner. No strategy for Player I can uniformly produce a well-order code at least as long as every code produced by Player II.
Every analytic set contained in has bounded rank: if is analytic, thenIn particular, the well-order codes produced continuously from all counterplays against one strategy have bounded ranks whenever they are all well-founded.
For a fibration over , the only possible differential in the homological Serre spectral sequence is . Identifying its two nonzero columns with the homology of the fiber gives the Wang homomorphism .
A Serre fibration is a map having the homotopy lifting property with respect to every disc, equivalently every CW complex.
A finite-length semisimple module has a canonical decompositionin which is the sum of all simple submodules isomorphic to a fixed simple module . Each is isomorphic to a finite direct sum .
The decomposition is intrinsic to the semisimple module: the isomorphism types of its simple submodules determine the summands.
The Page–Geilker experiment used a quantum random decision to choose a macroscopic source-mass configuration and measured its field with a torsion balance. The observed field followed the configuration in the observed branch rather than the nearly cancelling expectation-value average predicted by the simplest no-collapse semiclassical Einstein equation.
A biproduct is simultaneously a product in a category and a coproduct in a category, with projections and injections satisfying
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