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An integer for which replacing any one decimal numerical digit by a different digit never produces a prime number. The convention must specify whether changes creating a leading zero are permitted. An exact test enumerates the allowed substitutions and applies primality testing to each result.
For a sphere translating through an unbounded inviscid incompressible fluid at rest far away, the laboratory fluid velocity written in coordinates centred on the moving sphere is the gradient of the displayed dipole potential. Its radial velocity at the surface equals . The sphere-rest-frame velocity instead needs the additional uniform potential .
A piecewise constant barrier of finite width. Exact scattering follows by matching the wavefunction and its derivative at its two interfaces; below its height the internal wave number is imaginary.
A small sinusoidal relief on an incompressible Newtonian viscous half-space relaxes exponentially with the displayed timescale in the ideal free-surface model. Here is wavelength, not automatically an ice-sheet radius or diameter. Ice-history, finite mantle layering, elastic lithosphere and self-gravity complicate real glacial isostatic adjustment.
Enzymatic breakdown of cells by their own components after loss of normal regulation.
A reinforcing circuit in which two activities inhibit one another, potentially stabilizing mutually exclusive states.
For example, if A inhibits B and B inhibits A, a high-A, low-B state can reinforce itself because little B remains to oppose A. Multiple stable states still require suitable nonlinear response strengths; two inhibitory arrows alone are not a proof of bistability. [ref]
Molecular Biology of the Cell, 4th edition by Bruce Alberts and others. Chapters 3, 6 and 15, protein function, translation and cell signalling
Increasing genetic differentiation with geographic separation when gene flow tends to occur over shorter distances.
When most dispersal is local, nearby biological populations exchange more ancestry than distant ones, allowing genetic drift to generate a spatial pattern. The relationship is a population-level tendency rather than a rule that every pair must be ordered exactly by distance. Barriers and recent gene flow can modify it. [ref]
A model in which one functionally defined progenitor class can explain renewal through stochastic, balanced fates. Molecular uniformity and exact independence of neighbouring cells are stronger claims than the model requires.
A stage-structured population model tracks distinct life stages, such as juveniles and adults, with separate reproduction, survival and transition terms. A matrix update describes the linear case; density-dependent reproduction gives a nonlinear discrete dynamical system.
polysaccharide biosynthesis assembles sugar units into a polymer. Activated donors supply units, while the transferases determine linkage, branching and the relation between synthesis and transport.
hemicellulose denotes a group of plant-wall polysaccharides distinct from cellulose. Their association with cellulose can limit access, and complementary degrading enzymes can improve deconstruction.
A tromino is a polyomino made of three unit squares joined along whole sides. Up to translation, rotation and reflection, its two shapes are the straight three-square strip and the L shape. Their area is three, so a board tiled by trominoes must have area divisible by three; that condition alone need not ensure a board tiling. For example, a rectangle admits two straight trominoes.
Polynomial multiplication combines coefficients by convolution. If and , then the coefficient of in is .
For example, . The coefficient rule follows from the distributive property and . The school method takes coefficient operations for coefficients; Karatsuba multiplication and Fast Fourier transforms give faster methods for large inputs. Over exact coefficient rings, transform methods must still recover exact coefficients, rather than accepting uncontrolled rounding.
A polynomial greatest common divisor of two polynomials, not both zero, over a field divides both and is divisible by every common polynomial divisor. It is unique after requiring it to be monic, and the polynomial Euclidean algorithm computes it.
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