If only I knew how to count

November 19, 2008 at 1:43 pm | In Mathematics | 2 Comments

A question that held up my reading of a paper this week.

Let m be a squarefree integer. I could prove that something was bounded by k^r m^2 where m has r distinct prime factors. The paper had it bounded by d_k(m) m^2 where d_k(m) is the number of ways m can be written as a product of k factors.

Your exercise: prove these are the same. Combinatorists are welcome to laugh at me.

Elliptic curves talk

November 12, 2008 at 5:06 pm | In Mathematics, Uni | 4 Comments

My talk today was pretty well received. Talk linked as well as some calculations in Sage. If you saw my honours talk at UQ you might recognize some of the slides: this talk is better than that one though.

An unordered list

November 8, 2008 at 5:46 pm | In music | 3 Comments

5 favourite albums of 2008

Okkervil River – The Stand Ins
Favourite Self-Referential ennui: Bruce Wayne Campbell Interview – “Sick with singing. Singing the same song.”

The Lucksmiths – First Frost
Favourite Ridiculous Song Title: The National Mitten Registry

Gin Club – Junk
Favourite member: Conor McDonald. You can’t go wrong with this shambolic, gravel voiced man child.

You Am I – Dilletantes
Favourite Promotional Interview: Tim Rogers – “Rather than the meat and potatoes kneetrembler up against the school lockers, this one is more, gee, ah something involving a jus”.

Silver Jews – American Water (actually from 1998 but I only listened to it recently)
Favourite line: Random Rules – “Before we go I’ve got to ask you about that tan line on your ring finger”.

Elliptic curves, elliptic integrals, elliptic functions and ellipses

November 8, 2008 at 4:16 pm | In Mathematics | 2 Comments

In a moment of weakness I agreed to give a talk in the grad student colloquium next week. So I’m giving a friendly introduction to elliptic curves.

One thing I thought would be nice is to explain where the name comes from. So here’s the result of my web searching today. Any comments on whether this makes sense are welcome.

An ellipse has the equation x^2/a^2 + y^2/b^2 = 1, eccentricity e = \sqrt{1-b^2/a^2} and can be parametrised by x=a\cos\theta and y=\sin\theta.

To find the arc length of an ellipse from \theta = 0 to \phi we take the integral \int_0^{\phi} \sqrt{(dx/d\theta)^2+(dy/d\theta)^2} = \int_0^{\phi} \sqrt{a^2 \sin^2\theta + b^2 \cos^2\theta}. This is equal to \int_0^{\phi} \sqrt{b^2-(b^2-a^2)\sin^2\theta} = b\int_0^{\phi} \sqrt{1-(1-a^2/b^2)\sin^2\theta}. So call E(\phi,k) = \int_0^{\phi} \sqrt{1-k^2 \sin^2\theta} d\theta an incomplete elliptic integral of the second kind and the arc length is given by b E(\phi,\sqrt{1-a^2/b^2}).

Then people played around with these kind of integrals and came up with a few different kinds. Incomplete integrals of the first kind are of the form
F(\phi,k)= \int_0^{\phi} \frac{d\theta}{\sqrt{1-k^2\sin^2\theta}}.

This integral is kind of like the definition of inverse trigonometric functions by integrals. So somebody had the idea to invert them: Jacobi’s elliptic functions are defined by fixing k and finding an inverse to u = F(\phi,k) which allows us to define
\mathrm{sn} \; u = \sin \phi, \mathrm{cn} \; u = \cos \phi and \mathrm{dn} \; u = \sqrt{1-k^2\sin^2 \phi}. These are periodic along the real axis (because they are sin and cosine functions) but also have another period along the imaginary axis (I think, don’t really understand why).

We have the equations \mathrm{sn}^2 + \mathrm{cn}^2 = 1 and \mathrm{dn}^2 + k^2 \mathrm{sn}^2 = 1. On the intersection of these two surfaces we have \mathrm{dn}^2 = 1 - k^2 \mathrm{sn}^2 = 1 - k^2 (1-\mathrm{cn}^2)^2. But if we let y=\mathrm{dn} and x=\mathrm{cn} we have y^2 = \mathrm{quartic\ in } \ x which is an elliptic curve (if we change coordinates we can get the standard y^2 = x^3 + ax + b).

It turns out that elliptic curves can all be parametrised like this by elliptic functions (usually we use Weierstrass \wp-functions). So that’s the reason for the name: elliptic curves are parametrised by elliptic functions, which are inverses of elliptic integrals, which look like integrals that measure arc length of ellipses.

Baracking for Obama

November 4, 2008 at 8:42 am | In Politics | 2 Comments

(that joke wouldn’t work in this country… it’s a pity).

After avoiding the election for the past 20 months I’m kind of excited about it today. I wonder if I should tell my students to vote for Obama? Damn liberal professors…

If you’re waiting you might enjoy this song from The Mountain Goats which will remind you that all politicians are vampires anyway.

Stanford Twenty20 and the American market

November 1, 2008 at 6:02 pm | In Cricket, Life in America | 2 Comments

I read on Cricinfo that the Stanford matches are part of his plan to crack the American market. Well I can confirm that it hasn’t happened yet.

I watched the Stanford game via the internet channel ESPN360 – ESPN’s way to play things that they have the rights to but don’t have room for on the four tv channels that they own. Maybe a couple of the obscure-college-football fans that use ESPN360 clicked on the cricket link but I doubt it.

Still, it certainly will change the lives of some West Indian cricketers. A million dollars US is a lot of money for a twenty year old kid from Grenada.

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