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A parallelizable manifold is a smooth -manifold whose tangent bundle is trivial, equivalently one admitting smooth vector fields that form a basis of every tangent space. Every Lie group is parallelizable because a basis at the identity extends to a global frame by left translation.
A bump function is a smooth function with compact support. Rescaled and translated bump functions provide local cutoffs.
A smooth cutoff function is a smooth function used to localize an argument, typically equal to one on one region and zero outside a slightly larger region. Such functions can be built from bump functions and smooth partitions of unity.
Suppose a homogeneous scalar field oscillates much faster than the Hubble parameter in a monomial potential . The virial theorem averaged over one oscillation gives , and henceA quadratic minimum therefore behaves like pressureless matter, whereas a quartic minimum behaves like radiation in cosmology.
Starobinsky inflation is an inflationary model whose Einstein-frame scalar potential can be writtenThe exponentially flat large-field plateau supports slow-roll inflation, while the potential is quadratic near its minimum.
Single-field slow-roll inflation uses one canonical scalar clock whose potential evolves slowly enough for the slow-roll parameters to remain small. Its adiabatic long mode produces the single-field inflation squeezed-limit consistency relation.
The neutral pressure level is the height at which indoor and outdoor hydrostatic pressures agree. Flow through an opening reverses direction across this level.
For a walk with increments with probability and with probability , putThen is a martingale. Optional stopping at the first linear-boundary crossing and optimization over gives exponential maximal-deviation bounds governed by the Legendre transform of a cumulant-generating function.
Hindered settling is the reduction or modification of particle settling by the displaced carrier-fluid backflow and by interactions with other particles. A simple dilute mixture model gives every species the same volume-averaged backflow.
The Stokes settling velocity is the terminal relative velocity obtained by balancing gravity, buoyancy, and the Stokes drag law for a dilute isolated sphere.
The incidence graph of a set family is the bipartite graph whose two vertex classes are the family members and the ground points, with an edge for each containment relation.
The Boolean lattice is the power set of a finite set ordered by inclusion. Its rank- level consists of all -element subsets.
An entanglement witness is a Hermitian operator such that for every separable quantum state , but for at least one entangled state .
A partially linear model combines a linear coefficient with an unrestricted nuisance function, for example .
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