A Riemann P-symbol specifies a second-order Fuchsian equation with regular singular points and the two local exponents shown beneath each point. The Fuchs relation is .
Linearizing at extinction gives . The principal eigenfunction for Neumann data at and Dirichlet data at is , with growth rate
A population is viable exactly when this is positive, requiring and . Thus intrinsic growth must beat predation and the refuge must exceed the diffusion-dependent critical size.
The terms are logistic birth and crowding, , linear predation mortality , and diffusion . The coast is reflecting, so ; lethal habitat beyond gives the absorbing condition .
A word containing neither nor cannot change symbol. The language is therefore , hence is regular.
This language is not regular. Intersecting with the regular language isolates strings whose two zero blocks must have equal length; the pumping lemma then gives a contradiction.
The pumping lemma for regular languages says that sufficiently long accepted words split as with , , and accepted for all . Applying it to pumps only zeros, so language (i) is not regular.
After multiplying the five-point equations by if necessary, their matrix is symmetric positive definite. For any ordering write , where . Gauss-Seidel method uses and has iteration matrix . In part (a) take ; then and
which is positive definite. Therefore the iteration matrix has spectral radius below one for every ordering, proving convergence.
Let and suppose , so . Put and . Since , . Also
Positive definiteness makes this positive, hence for every eigenvalue and .
Tangential velocity is unchanged, while the downstream normal velocity is . Thus the incident and outgoing angles to the shock have tangents and . Using the tangent-difference identity, the ratio from part (b), and gives
Insert , eliminate and the downstream normal speed using mass conservation, and solve for the density ratio. This gives
In the shock frame, conservation of mass, normal momentum, and energy gives
These are the Rankine-Hugoniot relations.
Integrating the viscous traction over the side and end surfaces gives
It is always negative. In steady state the cylinder stores no angular momentum, so the divergence theorem for angular-momentum balance requires viscous torque to cancel net advective influx.
Adding inward side flux and outward end-cap flux gives the net inward angular-momentum advection
It is positive because , and tends to the stated value.
The only vorticity component is . Its steady equation is , solved by . Regularity at zero gives
Its maximum satisfies , , so .
The radial strain draws fluid inward, axial strain expels it away from the midplane, and supplies swirl. In cylindrical coordinates , so the flow is incompressible.

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!
We have two killer features:
  1. 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-calculus
    Articles 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/derivative
  2. 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.
    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.
  3. https://raw.githubusercontent.com/ourbigbook/ourbigbook-media/master/feature/x/hilbert-space-arrow.png
  4. Infinitely deep tables of contents:
    Figure 6.
    Dynamic article tree with infinitely deep table of contents
    .
    Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade
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