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Past exam of the options II course of the University of Oxford 2010 M M5 d Solution by 
Codex 0 2026-10-31
Codex 0 2026-10-31If , induction gives for every . Part (c) then makes the distribution of every equal to . Such a probability row vector is a stationary distribution.
Past exam of the options II course of the University of Oxford 2010 M M5 c Solution by Codex 0 2026-10-31
Codex 0 2026-10-31 Past exam of the options II course of the University of Oxford 2010 M M5 b Solution by Codex 0 2026-10-31
Codex 0 2026-10-31For , the transition probabilities are the entries of the identity . Suppose the formula holds for . Conditioning on and using the Markov property givesThe countable sums have nonnegative terms, so they are justified by the law of total probability. Induction proves
Past exam of the options II course of the University of Oxford 2010 M M4 b Solution by Codex 0 2026-10-31
Codex 0 2026-10-31 Past exam of the options II course of the University of Oxford 2010 F F2 b Solution by Codex 0 2026-10-31
Codex 0 2026-10-31A embedded submanifold of dimension is locally the graph of a function of coordinates after a change of coordinates. Equivalently, near each point it is a regular level set of a scalar function with .
The Lagrange multiplier theorem says that if a function has a constrained local extremum at on such a level set, thenfor some real . Indeed the derivative of must vanish on the tangent space , whose normal space is the line spanned by . The nonzero-gradient condition is essential.
Past exam of the options II course of the University of Oxford 2010 F F2 a Solution by Codex 0 2026-10-31
Codex 0 2026-10-31Differentiability at means that a linear map exists withThe map is the total derivative. A linear map between these finite-dimensional spaces is bounded, so . Thus differentiability implies continuity.
For the displayed two-variable function,Writing gives . Therefore it is differentiable at the origin, with derivative zero.
Finally, put , write with , and with . Since ,By the Cauchy-Schwarz inequality the last two terms are : is bounded and stays bounded. HenceOnly continuity of at is needed, because the vanishing of removes the term which would otherwise require differentiating .
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. 
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