For an infinitesimal test mass on the surface of a homogeneous sphere, its Newtonian mechanical energy is
The spatially flat case corresponds to zero specific energy. Using
and dividing by gives
Since is proportional to the scale factor , this is the Newtonian derivation of the flat Friedmann equation
Now take and . Put
The contracting branch of the Friedmann equation is
Integrating from gives
Therefore
on the contracting interval, reaching at .
In this discrete age-structured population model, individuals of age present during the breeding season each produce newborns, so the number entering age zero is
An age- individual survives the following winter with probability and is then age in the next year, giving
Together with the initial age distribution, these equations determine every later cohort.
Substitute into the survival equation. Then
and iteration gives
The birth equation becomes
A nonzero population mode therefore requires the Discrete Euler-Lotka equation
For nonnegative vital rates with at least one surviving reproductive age and enough decay to make the series finite, is continuous and strictly decreasing on , tends to infinity as , and tends to zero as . The intermediate value theorem gives one positive solution, and strict monotonicity makes it unique.
The value is the expected lifetime offspring count under these fixed rates. Since decreases, places the root at and the population grows geometrically; places it at and the population shrinks. Equality gives replacement, .
An exponential dispersion family has densities of the form
where is the canonical parameter, is the dispersion parameter, and the support does not depend on . Its mean and variance are
For the given Gamma distribution, set
Then
gives
Thus the gamma laws form the Gamma exponential dispersion family.
Directly integrating, or using the cumulant-generating function of a gamma distribution,
so
Differentiation at zero yields

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