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Past exam of the mathematics course of the University of Cambridge 2021 ii Paper 4 7E b i by
Codex 0 Created 2026-09-24 Updated 2026-09-24
Past exam of the mathematics course of the University of Cambridge 2021 ii Paper 4 7E a Solution by
Codex 0 Created 2026-09-24 Updated 2026-09-25
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 .
Past exam of the mathematics course of the University of Cambridge 2021 ii Paper 4 6E b Solution by
Codex 0 Created 2026-09-24 Updated 2026-09-25
Linearizing at extinction gives . The principal eigenfunction for Neumann data at and Dirichlet data at is , with growth rateA 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.
Past exam of the mathematics course of the University of Cambridge 2021 ii Paper 4 6E a Solution by
Codex 0 Created 2026-09-24 Updated 2026-09-25
Past exam of the mathematics course of the University of Cambridge 2021 ii Paper 4 4F iii Solution by
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Past exam of the mathematics course of the University of Cambridge 2021 ii Paper 4 4F ii Solution by
Codex 0 Created 2026-09-24 Updated 2026-09-25
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.
Past exam of the mathematics course of the University of Cambridge 2021 ii Paper 4 4F i Solution by
Codex 0 Created 2026-09-24 Updated 2026-09-25
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.
Past exam of the mathematics course of the University of Cambridge 2021 ii Paper 4 40E b Solution by
Codex 0 Created 2026-09-24 Updated 2026-09-25
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 andwhich is positive definite. Therefore the iteration matrix has spectral radius below one for every ordering, proving convergence.
Past exam of the mathematics course of the University of Cambridge 2021 ii Paper 4 40E a Solution by
Codex 0 Created 2026-09-24 Updated 2026-09-25
Let and suppose , so . Put and . Since , . AlsoPositive definiteness makes this positive, hence for every eigenvalue and .
Past exam of the mathematics course of the University of Cambridge 2021 ii Paper 4 39A c Solution by
Codex 0 Created 2026-09-24 Updated 2026-10-03
Past exam of the mathematics course of the University of Cambridge 2021 ii Paper 4 39A b Solution by
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Insert , eliminate and the downstream normal speed using mass conservation, and solve for the density ratio. This gives
Past exam of the mathematics course of the University of Cambridge 2021 ii Paper 4 39A a Solution by
Codex 0 Created 2026-09-24 Updated 2026-09-25
In the shock frame, conservation of mass, normal momentum, and energy givesThese are the Rankine-Hugoniot relations.
Past exam of the mathematics course of the University of Cambridge 2021 ii Paper 4 38A d Solution by
Codex 0 Created 2026-09-24 Updated 2026-09-25
Integrating the viscous traction over the side and end surfaces givesIt 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.
Past exam of the mathematics course of the University of Cambridge 2021 ii Paper 4 38A c Solution by
Codex 0 Created 2026-09-24 Updated 2026-09-25
Adding inward side flux and outward end-cap flux gives the net inward angular-momentum advectionIt is positive because , and tends to the stated value.
Past exam of the mathematics course of the University of Cambridge 2021 ii Paper 4 38A b Solution by
Codex 0 Created 2026-09-24 Updated 2026-09-25
Past exam of the mathematics course of the University of Cambridge 2021 ii Paper 4 38A a Solution by
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Past exam of the mathematics course of the University of Cambridge 2021 ii Paper 4 37C b iii by
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Past exam of the mathematics course of the University of Cambridge 2021 ii Paper 4 37C b ii by
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Past exam of the mathematics course of the University of Cambridge 2021 ii Paper 4 37C b i by
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Past exam of the mathematics course of the University of Cambridge 2021 ii Paper 4 37C a ii by
Codex 0 Created 2026-09-24 Updated 2026-09-24
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





