Mathematical_summary on Patrick Stevens

The tiny proof that primes 1 mod 4 are sums of two squares

Thu, 28 Sep 2023 00:15:00 +0000

Exploding the incredibly terse proof into a bunch of exposition.

The uncountability of the reals (a note from Hacker News)

Sun, 02 Aug 2020 00:00:00 +0000

A quick note from Hacker News about a beautiful proof of the uncountability of the reals.

Part III essay

Wed, 15 Jun 2016 00:00:00 +0000

Now that my time in Part III is over, I feel justified in releasing my essay, which is on the subject of Non-standard Analysis. It was supervised by Dr Thomas Forster (to whom I owe many thanks for exposing me to such an interesting subject, and for agreeing to supervise the essay).

Finitistic reducibility

Wed, 25 May 2016 00:00:00 +0000

A quick overview of the definition of the mathematical concept of finitistic reducibility.

Tennenbaum's theorem

Wed, 27 Apr 2016 00:00:00 +0000

Most recent exposition: an article on Tennenbaum’s Theorem. Comments welcome. The proof is cribbed from Dr Thomas Forster, but his notes only sketched the fairly crucial last step, on account of the notes not yet being complete.

Modular machines

Thu, 21 Apr 2016 00:00:00 +0000

I’ve written a blurb about what a modular machine is (namely, another Turing-equivalent form of computing machine), and how a Turing machine may be simulated in one. (In fact, that blurb now contains an overview of how we may use modular machines to produce a group with insoluble word problem, and how to use them to embed a recursively presented group into a finitely presented one.)

A modular machine is like a slightly more complicated version of a Turing machine, but it has the advantage that it is easier to embed a modular machine into a group than it is to embed a Turing machine directly into a group. We can use this embedding to show that there is a group with unsolvable word problem: solving the word problem would correspond to determining whether a certain Turing machine halted.

Independence of the Axiom of Choice (for programmers)

Wed, 13 Apr 2016 00:00:00 +0000

So you’ve heard that the Axiom of Choice is magical and special and unprovable and independent of set theory, and you’re here to work out what that means.

Another Monty Hall explanation

Fri, 08 Apr 2016 00:00:00 +0000

Recall the Monty Hall problem: the host, Monty Hall, shows you three doors, named A, B and C. You are assured that behind one of the doors is a car, and behind the two others there is a goat each. You want the car. You pick a door, and Monty Hall opens one of the two doors you didn’t pick that he knows contains a goat. He offers you the chance to switch guesses from the door you first picked to the one remaining door. Should you switch or stick?

Friedberg-Muchnik theorem

Fri, 05 Feb 2016 00:00:00 +0000

Another short post to point out my new article on the Friedberg-Muchnik theorem, a theorem from computability theory. It uses what is known officially as a finite injury priority method, and the proof is cribbed entirely from Dr Thomas Forster.

Multiplicative determinant

Fri, 01 Jan 2016 00:00:00 +0000

I’m clearing out my desktop again, and found this document on the multiplicativity of the determinant, which I wrote in 2014. It might as well be up here.

I should note that this document contains no motivation of any kind. It is simply an exercise in symbol-shunting, and it has no clever ideas in it.

My First Forcing

Sat, 28 Nov 2015 00:00:00 +0000

In the Part III Topics in Set Theory course, we have used forcing to show the consistency of the Continuum Hypothesis, and we are about to show the consistency of its negation. I don’t really grok forcing at the moment, so I thought I would go through an example.

Lottery odds

Fri, 25 Sep 2015 00:00:00 +0000

It has been proposed to me that if one is to play the National Lottery, one should be sure to select one’s own numbers instead of allowing the machine to select them for you. This is not an optimal strategy.

Proof by contradiction

Fri, 21 Aug 2015 00:00:00 +0000

Here I explain proof by contradiction so that anyone who has ever done a sudoku and seen algebra may understand it.

Matrix puzzle

Fri, 19 Dec 2014 00:00:00 +0000

I recently saw a problem from an Indian maths olympiad:

There is a square arrangement made out of n elements on each side (n^2 elements total). You can put assign a value of +1 or -1 to any element. A function f is defined as the sum of the products of the elements of each row, over all rows and g is defined as the sum of the product of elements of each column, over all columns. Prove that, for n being an odd number, f(x)+g(x) can never be 0.

Sum-of-two-squares theorem

Tue, 09 Sep 2014 00:00:00 +0000

*Wherein I detail the most beautiful proof of a theorem I’ve ever seen, in a bite-size form suitable for an Anki deck.

Statement

There’s no particularly nice way to motivate this in this context, I’m afraid, so we’ll just dive in. I have found this method extremely hard to motivate - a few of the steps are a glorious magic.

\(n\) is a sum of two squares iff in the prime factorisation of \(n\), primes 3 mod 4 appear only to even powers.

Proof

We’re going to need a few background results.

Solvability of nonograms

Sun, 13 Jul 2014 00:00:00 +0000

Recently, a friend re-introduced me to the joys of the nonogram (variously known as “hanjie” or “griddler”). I was first shown these about ten years ago, I think, because they appeared in The Times. When The Times stopped printing them, I forgot about them for a long time, until two years ago, or thereabouts, I tried these on a website. I find the process much more satisfying on paper with a pencil than on computer, so I gave them up again and forgot about them again.

Proof that symmetric matrices are diagonalisable

Mon, 26 May 2014 00:00:00 +0000

This comes up quite frequently, but I’ve been stuck for an easy memory-friendly way to do this. I trawled through the 1A Vectors and Matrices course notes, and found the following mechanical proof. (It’s not a discovery-proof - I looked it up.)

Lemma

Let \(A\) be a symmetric matrix. Then any eigenvectors corresponding to different eigenvalues are orthonormal. (This is a very standard fact that is probably hammered very hard into your head if you have ever studied maths post-secondary-school.) The proof of this is of the “write it down, and you can’t help proving it” variety:

Discovering a proof of Sylvester's Law of Inertia

Sat, 03 May 2014 00:00:00 +0000

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

Sat, 26 Apr 2014 00:00:00 +0000

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

Thu, 17 Apr 2014 00:00:00 +0000

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 \(B\)). Then consider how \(A-\lambda I\) acts on the vectors of \(B\); in particular, how it deals with the subspace spanned by \(b_1, \dots, b_i\).

Sample topology question

Tue, 15 Apr 2014 00:00:00 +0000

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.

Discovering a proof of Heine-Borel

Fri, 04 Apr 2014 00:00:00 +0000

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. Here, then, is my attempt to discover a proof of the theorem, using as a guide all the results I know but can’t necessarily prove without lots of effort.

How to discover the Contraction Mapping Theorem

Sun, 30 Mar 2014 00:00:00 +0000

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).

Three explanations of the Monty Hall Problem

Sun, 22 Dec 2013 00:00:00 +0000

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. He shows you three doors; behind one of the three is a car, and the other two hide empty rooms. You have a free choice: you pick one of the doors. Monty Hall then opens a door which you did not pick, which he knows is an empty-room door. Then he gives you the choice: out of the two doors remaining, you may switch your choice to the other door, or stick with the one you first picked. You will get whatever is behind the door you end up with. You want to pick the car; do you stick with your first choice, or do you switch to the other door?

Markov Chain card trick

Tue, 12 Nov 2013 00:00:00 +0000

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. Let \(P\) be randomly chosen from the range \(1 \dots 5\), and let \(X_0 = P\). Proceed as follows: define \(X_n\) as “the value of the \(\sum_{i=0}^{n-1} X_i\)-th card”. Stop when the newest \(X_n\) would be greater than \(52\).

How to do Analysis questions

Thu, 24 Oct 2013 00:00:00 +0000

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. For instance, one question from the first example sheet of Analysis II this year is as follows:

Slightly silly Sylow pseudo-sonnets

Sat, 31 Aug 2013 00:00:00 +0000

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.

Monorhymic Motivation ¹

Suppose we have a finite group called \(G\).
This group has size \(m\) times a power of \(p\).
We choose \(m\) to have coprimality:
the power of \(p\)’s the biggest we can see.
Then One: a subgroup of that size do we
assert exists. And Two: such subgroups be
all conjugate. And \(m\)’s nought mod \(n_p\),
while \(n_p = 1 \pmod{p}\); that’s Three.

Topology made simple

Mon, 26 Aug 2013 00:00:00 +0000

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 \(\tau\). (That is: \(\tau\) contains the empty set and the entire set; sets in \(\tau\) are subsets of \(X\); not-necessarily-countable unions of sets in \(\tau\) are in \(\tau\); and finite intersections of sets in \(\tau\) are in \(\tau\).) We then say that \((X, \tau)\) is a topological space.

The Orbit/Stabiliser Theorem

Mon, 22 Jul 2013 00:00:00 +0000

The Orbit/Stabiliser Theorem is a simple theorem in group theory. Thanks to Tim Gowers for the proof I outline here - I find it much more intuitive than the proof that was presented in lectures, and it involves equivalence relations (which I think are wonderful things).

Theorem: \(\vert {g(x), g \in G} \vert \times \vert {g \in G: g(x) = x} \vert = \vert G \vert\).

Proof: We fix an element \(x \in G\), and define two equivalence relations: \(g \sim h\) iff \(g(x) = h(x)\), and \(g \cdot h\) if \(h^{-1} g \in \text{Stab}_G(x)\), where \(\text{Stab}_G(k) = {g \in G: g(k) = k}\).

Sylow theorems

Wed, 26 Jun 2013 00:00:00 +0000

A fairly long and winding way through a proof of the three Sylow theorems.