ygn

Well, tomorrow.

This is some of Simon’s homework, because he was too lazy to do it himself and I was bored. I think it was proving that something was or wasn’t a field.

Anyway, as a new series of posts, I intend to post some results that I like. Some (such as this one) will be proven. If anyone has ideas for more recurring things I could blog (I have a few more) tell me.

So, today I’m going to prove a classic result. I think Euclid did it first, way back when people fought in the dark. Anyway.

Theorem: There are an infinite number of prime numbers.

Proof: Assume that there are only finitely many prime numbers, say $p_1, p_2, p_3, \dots, p_n$. Now let $q = p_1p_2p_3 \cdots p_n + 1$, and let $p$ be a prime factor of $q$, so that $q = p \cdot r$. If $p \in \{p_1, \dots, p_n\}$, then, without loss of generality, we can assume that $p = p_1$. Now,
\begin{align*}
q – p_1 \cdots p_n &= p \cdot r – p_1p_2 \cdots p_n \\
&= p_1 \cdot r – p_1p_2 \cdots p_n \\
&= p_1(r – p_2 \cdots p_n) \\
&= 1.
\end{align*}
Hence $p_1$ divides 1, which is total bollocks. Hence $p$ cannot be in the set $\{p_1, \dots, p_n\}$, and so our list of prime numbers is incomplete. $\qquad{\square}$

Also, writing $\LaTeX$ without my macros and tabs is hard. Though it is much easier now that I’ve made \$ work.

I’m going to try and explain what my problem is. And by problem, I mean what I’m working on. I have no idea what the audience knows, so I’m most probably going to assume an ``interesting’‘ level of mathematics. I’m also going to break it up into bits, because it makes it easier.

A golden-mean matroid, when viewed in two dimensions, is a collection of points and lines, such that the golden ratio is everywhere. That’s not a very good definition, but tough. The five-pointed star is a classic example, and there are others. This definition then generalises into higher dimensions (as many as you want, because mathematicians are awesome like that) and that’s where it starts to get interesting.

Now, for some more formal stuff.

A partial field is a field in which addition need not be defined.

For an example, consider the set \(\mathbb{U} = \{-1, 0, 1\} \). This is a partial field (known as the regular partial field). You can see that the set is closed under multiplication, and all the field axioms hold, except for closure under addition, as, for example, \( 1 + 1 = 2 \notin \mathbb{U} \).

The golden-mean partial field is \( \mathbb{G} = \{ \pm \phi^i \mid i \in \mathbb{Z} \}\), where \( \phi \) is the positive root of \( x^2 – x – 1 \).

A more useful characterisation of golden-mean matroids is as the intersection of \(GF(4)\) and \(GF(5)\) matroids. This is the one I’ve kinda been forced into using, as I don’t have unique representation of matroids over the golden-mean partial field, but I do over \(GF(4)\).

This is awesome.

\[
z \left( 1 \ +\ \sqrt{\omega_{i+1} + \zeta -\frac{x+1}{\Theta +1} y + 1}
\ \right)
\ \ \ =\ \ \ 1
\]

After those months of hiatus, I’m turning into a blogging machine. Maybe I finally have stuff to say?

Anyway, it’s my birthday sometime soon. And then Christmas sometime soon after that. So I shall make a list of what you should get me. This is for people like Melissa, who has as much imagination as a poet, and therefore need help. I’ll do my best to keep silly stuff out.

• Free Internet.
• One of those drink bottle backpacks.
• 2TB+ FW800 External HD.
• Rubiks Cubes with size sides strictly greater than 5.
• Books. Both for reading and graduate-level mathematics textbooks. Pester me if you want subject matter.
• Shorts.
• Tops. What I wear instead of jerseys.
• A pre-order of Diablo III.
• A pre-order of the special edition of Heart of the Swarm (when it becomes available).
• A desk lamp. That uses a stand (not a clip).
• A good quality multibox with at least 6 plugs.
• A desk organiser. For all my pens and stuff. Vertical storage (desk space is at a premium with speakers, laptop, and screen).
• A good quality bike.

There’s probably other stuff, but I forget. I’ll add it in if I remember at a good time (like when playing with Jaina).

As promised, I coerced Melissa into letting me put her talk on the Internet. Now you can all find out why she likes to study people with gas

This photo is stolen from a draft blog post that’ll never get finished. What song can relate to a Limnic Eruption anyway? Yes, I know it’s basically just the ground farting. So it’s not that hard. Then there’s Pyroclastic flow and Solar flares which are a little harder. As a bonus, those links all go to places that actually make sense. Which is different.

Anyway, I’ve spent the last few days in Sliverstream for the NZMASP conference, organised by Snook. As is customary at such conferences, I gave a talk. This one includes a very, very, very sketchy proof. And lots of lies and stuff. And a little bit of American folklore.

Anyway, Maximum-Sized Golden-Mean Matroids.

Also, Melissa won the Best Applied Maths talk, for her talk about people with GAS and stuff. I might coerce her into putting her slides up once she’s finished her shopping expedition to Palmy.

So, life. I find that photo quite appropriate.

I noticed that I haven’t blogged here in ages. Not as much ages as qwandor, but still ages. It could be that I haven’t done anything in that time, which is probably true. I’m a grad student. When do I ever do anything?

Anyway, I can’t really remember anyway. It has been ages. And Football Manager just crashed, which makes me grumpy. I have done some stuff that I might as well mention. This blog is all about me, after all. And I’ll make some general observations.

I went to see my sisters perform in 42nd Street. Got all dressed up (Snook and Ben had the best reactions) and found myself a date and everything. Oh, I wasn’t supposed to be there, so I shocked almost everyone at whatever half-time is called in the theatre when I stood up and made myself known. And then gave Melissa some flowers (so she wouldn’t feel bad about my date, which she didn’t anyway). They died. Oh, and the show wasn’t my thing. No explosions, lots of singing and dancing. Though it was probably the best evening I’ve had in a long time. Shows how sucky life is.

Oh yeah. Life still sucks.

Next, politics. If you’re a commie bastard, stop posting crap on the Internet, or I may be forced to unfollow/defriend you until all the hideous pictures of Annette King bugger off. Also, if you put crap into my letterbox I will not be voting for you. The worst offenders are the Greens. I wouldn’t vote for them anyway, but I take great pleasure in placing their spam either in the trash or the wind. Normally the wind wins. Also, NZ First and the MMP is Jesus movement get voted against. Kinda like how the A-Team got people to actually care about student body elections (also, yay for VSM. VUWSA can GGF at last.)

What else? Oh, my research is going along well. I might have a paper within a year. Depends how much of a nazi Dillon is going to be. And how lazy I am. And how easily the last few cases fall out. More on that later, hopefully.

So, in summary: I do basically nothing, life sucks, research is showing potential to actually work, and I don’t get musicals.

This shows why you should never let women shop.

So in a weekend or so, I am going to a family dinner at which we are going to abuse/freak out my sister’s boyfriend-with-the-name-of-a-girl (who happens to be one of those things that doesn’t eat meat). There are the obvious questions to ask, such as his views on the continuum hypothesis and why the hell he doesn’t eat steak. Oh, and pads vs. tampons, front-to-back, back-to-front, or side-to-side, and how awesome bacon is.

I also heard that this is a no-pants party.

Any other ideas for stuff to do? Remembering that my sister is stupid enough to invite him/it (real men eat meat) to a meal, knowing full well that me, Ivan, and Simon will be there…

This picture shows a knot decomposition.

Anyway, this is for me to put some useful stuff in. Mainly links (and even some to places only I can go), but whatever strikes my fancy at the moment.

I really should sort out the CSS to make the lists be normal lists. Too lazy though.

• Labels: get and put
• Detexify
• Random Verb
• Textile
• sudo /usr/libexec/locate.updatedb
• Random Letter

Code also goes all ick. I think I shall play with CSS now.