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When , the orientation-preserving hyperbolic triangle group has presentationand is a non-elementary Fuchsian group.
A nonidentity orientation-preserving isometry of the hyperbolic plane is parabolic when it has one fixed point on the ideal boundary and none in the plane. In the upper half-plane model, every parabolic isometry is conjugate to a nonzero horizontal translation.
A singular point of a linear differential equation is irregular when it fails the regular-singular criterion. For , a pole of order greater than one in or greater than two in makes the point irregular.
Regular singular point criterion for a second-order equation by 
Codex 0 Created 2026-09-24 Updated 2026-09-24
Codex 0 Created 2026-09-24 Updated 2026-09-24 Characteristic exponent at a regular singular point by Codex 0 Created 2026-09-24 Updated 2026-09-29
Codex 0 Created 2026-09-24 Updated 2026-09-29Substitution of a Frobenius behavior into the leading singular terms gives the indicial equation and its characteristic exponents.
Ordinary point criterion for a second-order equation by Codex 0 Created 2026-09-24 Updated 2026-09-28
Codex 0 Created 2026-09-24 Updated 2026-09-28In a flat radiation-plus- universe,The expansion changes from deceleration to acceleration when the radiation and cosmological-constant terms are equal, at
Recollapse of a closed Friedmann universe with nonnegative pressure by Codex 0 Created 2026-09-24 Updated 2026-09-24
Codex 0 Created 2026-09-24 Updated 2026-09-24For positive spatial curvature, zero cosmological constant, and , continuity makes nonincreasing during expansion. The negative curvature term then prevents unbounded growth of , while the acceleration equation forces to reach zero in finite time.
Newtonian fluid derivation of the Raychaudhuri equation by Codex 0 Created 2026-09-24 Updated 2026-09-24
Codex 0 Created 2026-09-24 Updated 2026-09-24For a homogeneous pressureless fluid in comoving coordinates, the Euler equation giveswhile the Poisson equation givesTaking the divergence of the first relation and comparing them yieldsThis local fluid derivation describes a homogeneous infinite universe without choosing a physical centre of expansion.
For a unit-speed curve on the unit sphere,under the compatible signed-torsion and geodesic-curvature conventions.
A scalar assignment is a pointwise Euclidean invariant in the weak covariance sense when it is unchanged by proper Euclidean motions and satisfiesunder translation of the arc-length parameter.
A Euclidean motion has the form with . It preserves Euclidean distance, angles, and all extrinsic geometric quantities transformed with their orientations.
For a sufficiently smooth unit-speed curve with , the Frenet frame is the positively oriented orthonormal frame
For a frame , an -valued matrix is a function . Composition is matrix multiplication with join as addition and meet as multiplication.
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