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So, from the 11th (which is this Friday coming up) until the 16th (the following Wednesday), Melissa will be off on an epic quest to the land of misspellings and easter eggs and I will be here, with next-to nothing to do (I do have 8 hours of marking that will probably take under 3 to actually do sometime between 1500 Friday and 1200 Monday, as well as needing to be at work from 0900 to 1000 on Monday and Wednesday).

Also, I have noticed, that with the exceptions of my family(s) and work people, I don’t get to see people much any more. Probably since I got married. Maybe a little after that. Honours at least.

Anyway, to rectify both these situations, I am proposing that people do stuff with me. There are a few issues (which are open to negotiation, depending on the awesomeness of whatever it is):

• I have no transport except my legs. So stuff like “can you meet me in Karori?” will get a “no”. However, I can get to town or Kelburn or someplace like that. Basically anyplace within about an hour of walking. In case you don’t know, I live in Kilbirnie.
• I have no spare money. So don’t try to take me shopping or something like that. Plus I don’t like shopping.
• I don’t get dressed up. If I can’t go somewhere in shorts, I’m not going.

So, a few suggestions:

• Make me cook you dinner (not sure how safe this will be).
• Go for a walk.
• Buy a ginormous bag of candy, and give it to me.
• Get a big red crayon, and practise writing “0” on lots of assignments.
• Make me play games (this will have the added bonus of annoying Melissa).
• Whatever it is that people do with their friends. Chill or whatever it is.
• Watch a movie (I have a big arse screen, and I could probably even get a projector (though not at my house)).
• Pretty much anything else you can think of (modulo those almost-conditions above).

Oh, how to contact me. If you don’t already know. I have a cellphone (021366055) that I will endeavour to actually look at. Just don’t be surprised if I ring as opposed to texting. I also have a lot of email addresses which are all of the form $\langle$some selection of characters$\rangle$ shift-2 yomcat.geek.nz. There are about 3 selections of characters that won’t get to me, and they’re very hand to figure out. Or you could click the email link over here.

I need a hero to save me now
I need a hero
I need a hero to save my life
A hero’ll save me

I have just completed my PhD proposal. When (if) it gets accepted, I will be moved from provisional to full registration.

1. Do I have nothing to say, or do I want to say nothing?
2. Why am I so opposed to organised religion?
3. Is it worth it to care?
4. Am I really going to be ready to submit by the end of the year?
5. How do I get rid of door-knocking salespeople, apart from working ``normal’‘ hours?
6. Will I make it to 100 in Football Manager?
7. What is the point of everything?
8. I appear to be mellowing out.
9. Are existing friendships worth cultivating, even if the effort does not appear to be symmetric?
10. Is explaining what I really do possible?
11. Will Melissa find employment, keeping me in Wellington?
12. Do I want to stay in Wellington?
13. Why do certain things rile me up so much?
14. How important is money?
15. How do engineers solve the rigid body problem?
16. Is life really better now, or is it just me thinking it is for some reason?
17. Why do I keep forgetting the random verb?
18. What can I do?
19. Is social media worth continuing?

This photo is taken from just outside Ivan’s room.

I’ll get to a real interesting theorem sometime. Hopefully. This one is pretty nifty. It’s even one of those Fundamental theorems. Like the one that basically says calculus works, bitches! Which I won’t get in to, because it’s dull.

Theorem: Let $n$ be a member of $\mathbb{Z}^+ \setminus \{1\}$. Then $n$ can be written as a unique product of prime numbers.

I’m not going to prove this. Because I can’t be bothered. It’s not that hard, being in Euclid and all that. Basically. Anyway, the important word to note is unique. There is some fiddling, as you can write $6 = 2 \cdot 3$ and $6 = 3 \cdot 2$ (yay for isomorphisms). Oh, another consequence of this is that 1 is not prime. Not that it ever was in the first place. Though some people seem to think that it is. If you find one, hit them with a large stick. Or be polite and say “Sir, I do believe you are mistaken”.

This is Rob’s attempt at a snowman, when we went wandering in the Ruahines

Today’s Theorem comes thanks to Simon, because he is stupid. It is given without proof, because I don’t know how to prove it, and my book doesn’t have a proof. I probably don’t even know what it says.

Theorem:
Suppose that a function $f$ is represented by a power series in $x – x_0$ that has a nonzero radius of convergence $R$; that is,
\[
f(x) = \sum_{k=0}^{\infty} c_k(x – x_0)^k \qquad (x_0 < x < x_0 + R).
\]

1. If the power series representation of $f$ is integrated term by term using an indefinite integral, then the resulting series has radius of converge $R$ and converges to $\int f(x) dx$ on the interval $(x_0 – R, x_0 + R)$; that is,
   \[
   \int f(x)dx = \sum_{k=0}^{\infty} \left[ \int c_k(x – x_0)^k dx \right] + C \qquad (x_0 – R < x < x_0 + R)
   \]
   
2. If $\alpha$ and $\beta$ are points in the interval $(x_0 – R, x_0 + R)$, and if the power series representation of $f$ is integrated term by term from $\alpha$ to $\beta$, then the resulting series of numbers converges absolutely on the interval $(x_0 – R, x_0 + R)$ and
   \[
   \int_{\alpha}^{\beta} f(x) dx = \sum_{k=0}^{\infty} \left[\int_{\alpha}^{\beta} c_k(x – x_0)^k dx \right]
   \]
   

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 \(GF(4)\).