Discovering a proof of Sylvester's Law of Inertia

This is part of what has become a series on discovering some fairly basic mathematical results, and/or discovering their proofs. It’s mostly intended so that I start finding the results intuitive - having once found a proof myself, I hope to be able to reproduce it without too much effort in the exam. Statement of the theorem Sylvester’s Law of Inertia states that given a quadratic form \(A\) on a real finite-dimensional vector space \(V\), there is a diagonal matrix \(D\), with entries \(( 1_1,1_2,\dots,1_p, -1_1, -1_2, \dots, -1_q, 0,0,\dots,0 )\), to which \(A\) is congruent; moreover, \(p\) and \(q\) are the same however we transform \(A\) into this diagonal form.

Sequentially compact iff compact

Prof Körner told us during the IB Metric and Topological Spaces course that the real meat of the course (indeed, its hardest theorem) was “a metric space is sequentially compact iff it is compact”. At the moment, all I remember of this result is that one direction requires Lebesgue’s lemma (whose statement I don’t remember) and that the other direction is quite easy. I’m going to try and discover a proof - I’ll be honest when I have to look things up.

Cayley-Hamilton theorem

This is to detail a much easier proof (at least, I find it so) of Cayley-Hamilton than the ones which appear on the Wikipedia page. It only applies in the case of complex vector spaces; most of the post is taken up with a proof of a lemma about complex matrices that is very useful in many contexts. The idea is as follows: given an arbitrary square matrix, upper-triangularise it (looking at it in basis \(B\)).

Sample topology question

As part of the recent series on how I approach maths problems, I give another one here (question 14 on the Maths Tripos IB 2007 paper 4). The question is: Show that a compact metric space has a countable dense subset. This is intuitively clear if we go by our favourite examples of metric spaces (namely \(\mathbb{R}^n\), the discrete metric and the indiscrete metric). Indeed, in \(\mathbb{R}^n\), which isn’t even compact, we have the rationals (so the theorem doesn’t give a necessary condition, only a sufficient one); in the indiscrete metric, any singleton \({x }\) is dense (since the only closed non-empty set is the whole space); in the discrete metric, where every set is open, we can’t possibly be compact unless the space is finite, so that’s why the theorem doesn’t hold for a topology with so many sets.

Useful conformal mappings

This post is to be a list of conformal mappings, so that I can get better at answering questions like “Find a conformal mapping from <this domain> to <this domain>”. The following Mathematica code is rough-and-ready, but it is designed to demonstrate where a given region goes under a given transformation. whereRegionGoes[f_, pred_, xrange_, yrange_] := whereRegionGoes[f, pred, xrange, yrange] = With[{xlist = Join[{x}, xrange], ylist = Join[{y}, yrange]}, ListPlot[ [email protected] Through[{Re, Im}[ f /@ (#[[1]] + #[[2]] I & /@ Select[Flatten[Table[{x, y}, xlist, ylist], 1], With[{z = #[[1]] + I #[[2]]}, pred[z]] &])]]]] Möbius maps - these are of the form \(z \mapsto \dfrac{az+b}{c z+d}\).

Discovering a proof of Heine-Borel

I’m running through my Analysis proofs, trying to work out which ones are genuinely hard and which follow straightforwardly from my general knowledge base. I don’t find the Heine-Borel Theorem “easy” enough that I can even forget its statement and still prove it (like [I can with the Contraction Mapping Theorem][2]), but it turns out to be easy in the sense that it follows simply from all the theorems I already know.

How to discover the Contraction Mapping Theorem

A little while ago I set myself the exercise of stating and proving the Contraction Mapping Theorem. It turned out that I mis-stated it in three different aspects (“contraction”, “non-empty” and “complete”), but I was able to correct the statement because there were several points in the proof where it was very natural to do a certain thing (and where that thing turned out to rely on a correct statement of the theorem).

A roundup of some board games

It has been commented to me that it’s quite hard to find out (on the Internet) what different games involve. For instance, Agricola is a game about farming (and that’s easy to find out), but what you actually do while playing it is not easy to discover. Here, then, is a brief overview of some games. Agricola Agricola is a game in which you control a farm, and are aiming to make your farm thrive.

Rage, rage against the poet’s hardest sell

I feel that I can write a sonnet well. While sonnets are an easy thing to spout, It’s really hard to write a villanelle. By rhyming, any story I can tell: in couplets, rhyme and rhythm evens out. I feel that I can write a sonnet well. But alternately-structured verse is hell. The poet struggles, juggles words about: It’s really hard to write a villanelle. Enthusiasm’s difficult to quell. An acolyte of Shakespeare, I’m devout:

Writing essays

The aim of this post is twofold: to find out whether a certain mental habit of mine is common, and to draw parallels between that habit and the writing of essays. I don’t know whether this is common or not, but when I’m feeling particularly not-alert (for instance, when I’m nearly asleep, or while I’m doing routine tasks like cooking), I sometimes accidentally latch onto a topic and mentally explain it to myself, as if I were teaching it to the Ancient Greeks (who, naturally, speak English).

Introduction to functional programming syntax of Mathematica

Recently, I was browsing the Wolfram Community forum, and I came across the following question: What are the symbols @, #, / in Mathematica? I remember that grasping the basics of functional programming took me quite a lot of mental effort (well worth it, I think!) so here is my attempt at a guide to the process. In Mathematica, there are only two things you can work with: the Symbol and the Atom.

Denouement of Myst III: Exile

A long time ago, in a galaxy far far away, I completed Myst III: Exile. It’s a stupendously good puzzle game. For some reason, it popped into my mind again a couple of days ago. This post contains very hefty spoilers for that game (it will completely ruin the ending - I will be discussing information-exchange protocols which are key to completing it), so if you’re ever going to play it, don’t read this post yet.

The Creation

Once upon a time, before this bountiful age of Matter and Light, there was only the Fell. A single being, surrounded by Chaos, content to remain alone forever (for it did not know what a “friend” was). It had not the power to shape the Chaos; neither had it the inclination, for it needed nothing and had no desires. For seething unchanging aeons, it persisted. Then Chaos bore new fruit. A single electron, a point source of charge.

Smartphone Charter

I am shortly to receive a new Nexus 5. I am determined not to become a smartphone zombie, and so I hereby commit to the following Charter. I will keep my phone free of social networking apps, and I will ensure that I do not know the passwords to access their web interfaces. While they can be really quite handy, they are usually simply a distraction. People are used to the fact that I am present on the Internet only when I have my computer with me; there’s no need for that to change.

Three explanations of the Monty Hall Problem

Earlier today, I had a rather depressing conversation with several people, in which it was revealed to me that many people will attempt to argue against the dictates of mathematical and empirical fact in the instance of the Monty Hall Problem. I present a version of the problem which is slightly simpler than the usual statement (I have replaced goats with empty rooms). Monty Hall is a game show presenter.

The Training Game

The book Don’t Shoot the Dog, by Karen Pryor, contains a simple exercise in demonstrating clicker training. This is a very successful technique used to produce behaviour in animals: having first associated the sound of a click with the reward of attention or food, one can then use the click as an immediate substitute for the reward (so that one can train more complicated, time-critical actions through positive reinforcement; a click is instant, but food or attention requires the trainer approaching the trainee).

The Jean-Paul Sartre Cookbook

Many thanks to the Guru Bursill-Hall for bringing this tract to my attention through his weekly History of Maths bulletins. It was originally written in 1987 by Marty Smith, according to the Internet. The Jean-Paul Sartre Cookbook October 3. Spoke with Camus today about my cookbook. Though he has never actually eaten, he gave me much encouragement. I rushed home immediately to begin work. How excited I am! I have begun my formula for a Denver omelet.

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.