Markov Chain card trick

In my latest lecture on Markov Chains in Part IB of the Mathematical Tripos, our lecturer showed us a very nice little application of the theorem that “if a discrete-time chain is aperiodic, irreducible and positive-recurrent, then there is an invariant distribution to which the chain tends as time increases”. In particular, let \(X\) be a Markov chain on a state space consisting of “the value of a card revealed from a deck of cards”, where aces count 1 and picture cards count 10.

My quest for a new phone

*This post is unfinished, and may never be finished - I have decided that the Nexus 5 is sufficiently cheap, nice-looking and future-proof to outweigh the boredom of continuing the research here, especially given that such research by necessity has a very short lifespan. I am one of those people who hates shopping with a fiery passion. * My current phone is a five-year-old Nokia 1680. It has recently developed a disturbing tendency to turn off when I’m not watching it.

How to do Analysis questions

This post is for posterity, made shortly after Dr Paul Russell lectured Analysis II in Part IB of the Maths Tripos at Cambridge. In particular, he demonstrated a way of doing certain basic questions. It may be useful to people who are only just starting the study of analysis and/or who are doing example sheets in it. The first example sheet of an Analysis course will usually be full of questions designed to get you up and running with the basic definitions.

The Ravenous

Once upon a midnight dreary, while I pondered, weak and weary, I required a snack to feed me. Reaching in the kitchen drawer - With the scissors, cut the wrapping, I revealed a jar of tapen- Ade of olives. Gently snapping, snapping off the lid, I saw: Lines of mouldy olive scored the tapenade. The lid I saw Speckled with each mocking spore. How the pangs of hunger rumbled while I cursed the jar I’d fumbled;

Training away mental bias

In which I recount an experiment I have been performing. Please be aware that in this article I am in “[meaning what I say][1]” mode. For the past year or so, I have been consciously trying to identify and counteract places in the “natural”, everyday use of language in which gender bias is implicitly assumed to be correct. The kind of thing I mean is: A: I called the plumber.

Meaning what you say

In conversation with (say, for the purposes of propagating a sterotype) humanities students, I am often struck by how imprecisely language is used, and how much confusion arises therefrom. A case in point: A: I think that froogles should be sprogged! B: Sprogging froogles would make the bimmers go plog. A: But I use froogles all the time - I don’t care about the bimmers! Why are you so caught up on the plogging of bimmers?

Plot Armour

Wherein I dabble in parodic fiction. The title refers to the TV Tropes page on Plot Armour, but don’t follow that link unless you first resolve not to click on any links on that page. TV Tropes is the hardest extant website from which to escape. Jim, third-in-command of the Watchers, ducked behind the Warlord’s force-field, desperately trying to catch his breath in the face of an inexorable onslaught. His attackers, the hundred-strong members of the Hourglass Collective, had never been defeated in pitched battle.

How to prove that you are a god

I came across an interesting question while reading the blog of Scott Aaronson today. The question was as follows: In the world of the colour-blind, how could I prove that I could see colour? I’m presuming, to make the discussion more life-like and less cheaty, that this civilisation hasn’t discovered that light comes in wavelengths, or that it has but it can’t distinguish very well between wavelengths (so that all coloured light falls into the same bucket of 100nm to 1000nm, for instance).

Stumbled across 14th September 2013

On the merits of silence (I wholeheartedly agree): Given the previous results on humans’ sense of physical location, I’m not particularly surprised that you can make yourself identify your body as being somewhere other than where it really is: Aaand the future arrives: Another reason why Finland is amazing: A thought-provoking story: WebCite version On the “mundane magics” kind of lines: http://i.

Slightly silly Sylow pseudo-sonnets

This is a collection of poems which together prove the Sylow theorems. Notes on pronunciation Pronounce \( \vert P \vert \) as “mod P”, \(a/b\) or \(\dfrac{a}{b}\) as “a on b”, and \(=\) as “equals”. \(a^b\) for positive integer \(b\) is pronounced “a to the b”. \(g^{-1}\) is pronounced “gee inverse”. “Sylow” is pronounced “see-lov”, for the purposes of these poems. \(p\) and \(P\) and \(n_p\) are different entities, so they’re allowed to rhyme.

Topology made simple

I’ve been learning some basic topology over the last couple of months, and it strikes me that there are some very confusing names for things. Here I present an approach that hopefully avoids confusing terminology. We define a topology \(\tau\) on a set \(X\) to be a collection of sets such that: for every pair of sets \(x,y \in \tau\), we have that \(x \cap y \in \tau\); \(\phi\) the empty set and \(X\) are both in \(\tau\); for every \(x \in \tau\) we have that \(x \subset X\); and that \(\displaystyle \cup_{\alpha} x_{\alpha}\) is in \(\tau\) if all the \(x_{\alpha}\) are in \(\tau\).

Stumbled across 24th August 2013

The much-vaunted Hyperloop looks really cool, if it could ever be built: But it may be a bit too half-baked: I love a good visualisation: I laughed pretty much constantly through this piece of bureaucracy-hacking: This is a problem with the Internet of Things as well as with mind-computer interfaces: Wow - it’s possible to represent words as vectors so that vector(‘Paris’) - vector(‘France’) + vector(‘Italy’) results in a vector that is very close to vector(‘Rome’): https://code.

How to punt in Cambridge

When in Cambridge… The river is always full of beginners and professional puntists. The beginners veer all over the place, getting very wet, while the professionals zip between them, somehow managing to avoid collision by the width of an otter’s hair. The worst attempt by a beginner I’ve ever seen at punting was an attempt to use the pole rather like an oar, without ever touching the bottom of the river with it.

My experiences with flow

I’m in the middle of reading Flow, by Mihály Csíkszentmihályi, and so far, I love it. It describes the “flow state” of consciousness, that state of “everything is irrelevant except for the task at hand” in which time flies past without your noticing, and you don’t notice hunger or thirst or people moving around you. Flow can be induced when performing a difficult task which lies within your abilities, where immediate feedback is provided.

Thinking styles

All the way back into primary school (ages 4 to 11 years old, in case a non-Brit is reading this), we have been told repeatedly that “people learn things in different ways”. There were two years in primary school when I had a teacher who was very into Six Thinking Hats (leading to the worst outbreak of headlice I’ve ever encountered) and mind maps. I never understood mind maps, and whenever we were told to create a mind map, I’d make mine as linear and boxy as possible, out of simple frustration with the pointless task of making a picture of something that I already had perfectly well-set-out in my mind.

Stumbled across 11th August 2013

A thousand times this: A possible fix for the “economic problem of democracy”: A fascinating look at privacy online, how we’re not built for privacy, and how tribal cultures attain privacy: I’m all for healthy competition and so forth, but do we really want such massive phones? This is the kind of thing that I never quite have the courage or the morals to do: http://www.

New computer setup

In case I ever have to get a new computer (or, indeed, in case anyone else is interested), I hereby present the (updating) list of applications and so forth that I would immediately install to get a computer up to usability. Browser: Firefox with Ghostery, HTTPS Everywhere, and NoScript (and remember to turn on Do Not Track…) Mail client: Thunderbird with Enigmail Messaging client: Adium on Mac, and possibly Pidgin for others - I’ve never used a non-Mac chat client.

Stumbled across 4th August 2013

An ad developer has misgivings: Hint for dealing with some automated phone helplines - swear at them and they’ll put you through to a human: The future is coming: A large collection of replacements for various PRISM-vulnerable services: Some people think in a really rather interesting way: The joys of a memoryless distribution: An impressive photograph: largest photo cached A fair chunk of the “1910’s predicted Year 2000 technologies” has been invented: http://www.

On to-do lists as direction in life

Getting Things Done has gathered something of a cult following [archived due to link rot] since its inception. As a way of getting things done, it’s pretty good - separate tasks out into small bits on your to-do list so that you have mental room free to consider the bigger picture. However, there’s a certain aspect of to-do lists that I’ve not really seen mentioned before, and which I find to be really helpful.

Stumbled across 29th July 2013

Hehe: Wow - light trapped for a full minute: The importance of a consistent utility function: Obama promised to be friendly to whistleblowers, and has quietly removed said promise: I wholeheartedly agree with this site: Good post on belief-in-belief: Huh. A strange system, the US medical system: Very much this - about how the media has lost the plot about PRISMgate: Aaand my faith in humanity is once again shattered: http://i.