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A subset of a Polish space has the perfect set property when it is countable or contains a nonempty perfect subset. Every uncountable analytic set contains a perfect subset and consequently has cardinality .
A pointclass is a collection of subsets of topological spaces, usually specified by a common definability or closure condition.
The set-theoretic Baire space is , the set of all infinite sequences of natural numbers, with the product topology obtained from discrete .
Stationarity of a static solution under Derrick scaling requires . This necessary relation among the separately scaling energy terms is the Derrick virial identity.
Under Derrick scaling, an energy term containing spatial derivatives and homogeneous of degree in those derivatives scales as in spatial dimensions.
Bourgain bound for three-term-progression-free sets by 
Codex 0 Created 2026-09-28 Updated 2026-10-03
Codex 0 Created 2026-09-28 Updated 2026-10-03If is a finite abelian group of odd order and contains no nonconstant three-term arithmetic progression, then Bourgain's bound is
If solves the trapped stationary equation, the rescaled quadratic-phase ansatzsolves the free mass-critical equation when the scaling, chirp, and phase parameters satisfy the associated lens-transform system.
To prove that a threshold function requires every input bit in the worst case, an adversary answers queries while keeping completions on both sides of the threshold possible. If this remains true until the final unqueried bit, every decision tree has depth equal to the number of variables.
A matrix is an -approximate inverse of a Gram matrix in entrywise maximum norm whenThis turns the bias remainder of a Debiased Lasso into a product of and the coefficient error.
The three special conformal transformations complete translations, rotations, and dilations to the ten-dimensional de Sitter isometry group in flat slicing. A propagation speed different from the metric light speed generally breaks these transformations.
In flat conformal coordinates, a de Sitter dilation acts as . It is an isometry because the common coordinate rescaling cancels the inverse-square scaling of the conformal factor.
A spatial translation in de Sitter flat slicing sends at fixed conformal time. Translation invariance of a momentum-space correlator produces its momentum-conserving Dirac delta distribution.
The concentration of a spherical dark-matter halo compares its outer radius with an inner scale radius. For a Navarro--Frenk--White profile, .
Suppose a porous layer has thickness and permeability of a porous medium , and the pressure drop acts across its thickness. Darcy's law gives the drainage volume fluxIf , then .
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