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For , , and every ,with analogous estimates for lower Hölder seminorms. Scaling reduces the result to the unit ball.
The integrated Sachs-Wolfe effect is the CMB temperature shift produced when the gravitational potentials traversed by a photon evolve with time. Its line-of-sight source is proportional to .
The Koebe distortion theorem gives universal upper and lower bounds on a univalent function and its derivative in terms of the distance to the boundary. In particular, derivatives at two points whose hyperbolic distance is bounded differ by at most a universal multiplicative factor.
For a reversible Markov chain with stationary flow , the conductance of a set isIts Cheeger constant minimizes the boundary flow divided by the smaller stationary mass of the two sides of the cut.
The -pseudospectrum consists of spectral points together with resolvent points where . Equivalently it is described by a small lower norm.
For a closed operator, the resolvent is operator-valued holomorphic on its resolvent set. On a Hilbert space its norm is subharmonic and therefore cannot have a strict interior maximum.
The critical surface is the stable manifold of a critical renormalization-group fixed point. Tuning every relevant direction places a theory on this surface.
Stability matrix of a renormalization-group fixed point by 
Codex 0 Created 2026-09-24 Updated 2026-09-28
Codex 0 Created 2026-09-24 Updated 2026-09-28The stability matrix is the Jacobian of the beta functions at a renormalization-group fixed point. Its eigenvalues classify perturbations as relevant, irrelevant, or marginal.
If is the augmentation ideal of , a free presentation gives an exact sequenceThe middle module is free over on the free generators of .
A sign-flip randomization test independently multiplies observations by signs in . It is finite-sample exact for a zero-centered null when the joint distribution is invariant under those sign changes.
A permutation test uses relabelings of observations as its null-preserving transformations and computes a p-value from the resulting orbit of the test statistic.
When , the Radon-Nikodym derivative is the almost-everywhere unique measurable function satisfying .
On a locally Noetherian scheme, a coherent sheaf is a quasi-coherent sheaf that is locally associated with a finitely generated module. Kernels, cokernels, and finite direct sums of coherent sheaves remain coherent.
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