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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.
A pure stationary state is dark when every jump operator acts on it as a scalar, after an allowed shift of the jump operators, and the corresponding effective Hamiltonian preserves its ray. Every trajectory that reaches such a state remains on that same ray.
On , a standard Brownian motion has pathwise quadratic variation , whereas the time-rescaled process has quadratic variation . Therefore their path laws are mutually singular when .
The quadratic covariation of two continuous semimartingales is the limit in probabilityThe polarization identity gives .
For a finite extension and all sufficiently large , the p-adic logarithm and p-adic exponential are inverse group isomorphisms
A state is a positive functional of norm one. The state space is a weak-star compact convex subset of .
For independent Poisson observations with fitted means , the deviance from the saturated model iswith .
The Pauli-Lubanski pseudovector is , up to an overall sign convention. Its square is a Casimir of the Poincare algebra and measures spin in a massive irreducible representation.
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