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A goodness-of-fit test compares a discrepancy statistic with its distribution under a fitted null model. A small p-value indicates that the observed discrepancy would be unusual if that model were correct.
The GHZ theorem uses perfect multipartite quantum correlations to contradict local predetermined values without a statistical inequality. For the three-qubit GHZ state, three certain Pauli-product relations force a fourth product to have the opposite sign from its quantum-mechanical value.
For a metric , the Lagrangian has the affinely parametrized geodesic equations as its Euler-Lagrange equations. Its value is half the constant squared norm of the tangent.
The generation-interval distribution gives the probability law of the generation interval. In discrete time, its probabilities form a nonnegative sequence summing to one.
Negative binomial regression models overdispersed counts with a negative binomial distribution, commonly using a logarithmic link for the mean.
Quasi-Poisson regression uses the mean model and variance relation . It estimates through quasi-likelihood and scales its covariance by the estimated dispersion .
The stationary exponential covariance function isThe corresponding Gaussian process is Markov and is a stationary Ornstein-Uhlenbeck process.
A centered continuous Gaussian process that is also a martingale has independent increments. Every future increment has zero covariance with every finite vector of past values by the martingale property; uncorrelated jointly normal variables are independent.
A functional linear model relates functional or scalar responses to functional predictors through a linear operator.
A covariance-operator distance compares positive trace-class operators while respecting either their linear embedding in operator space or a chosen factorization geometry.
Functional principal component analysis diagonalizes a compact covariance operator. Its eigenfunctions are principal component functions, and projecting a centered observation onto them gives uncorrelated functional principal component scores.
For fixed density , rank , and progression length, every sufficiently large proper coset progression of rank at most has the property that each subset of relative density at least contains a nontrivial arithmetic progression of that length. This follows from the multidimensional Szemerédi theorem.
For a free presentation , the relation module is the abelianization equipped with the -action induced by conjugation in .
Several essential spectra distinguish failure of upper and lower semi-Fredholm properties. For , these failures are detected at zero by the positive self-adjoint operators and .
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