nogods4me

Yo, e3pi, I've got 2 words for you: try decaf

NialScorva

Looks like freestyle mathematics-inspired poetry to me, and it's pecular flow strikes a chord with me.

Muad'Dib

Markov chains come up a lot in stats and linear algebra, so I know a little about them. I should probably actually read the link before pronouncing judgment on what it is and is not, though.

Muad'Dib

In my undergraduate research project I worked with elliptic curves, which are defined to be sets of complex ordered pairs (x,y) satisfying the relationship y^2 = f(x) = x^3 + ax^2 + bx + c, where a, b, and c are rational.

It's possible (though not pretty, without illustrations) to define an operation + on the points of an elliptic curve such that the points on the curve, together with a special point in the projective plane called "infinity", form a group.

If you're familiar with groups, I'd be happy to continue, but if you're new to the topic, this page cites all the algebra you'll need--though it doesn't really cover it in an intuitive or understandable way if you've not seen it before. I haven't ever seen a webpage that covers the idea behind groups from a beginner's perspective.

IMO, learning about groups (and rings) is way more useful than my esoteric research, which in many respects is just an application of certain branches of ring theory. It's fascinating, though, and if you get me started I may never stop!

wade-w

I'm not completey new to abstract algebra. Its a long story, but I knew what a group was before I'd ever seen the quadratic formula.

However, the link is still appreciated, since its been a long time. But please, continue! Or maybe we should take it to a new thread. I know there are a few other people who are interested in pure math here.

Muad'Dib

Once you know what groups are, it's not hard to explain the rest of what I did.

It turns out that the point at infinity functions as the identity element. Call this point O.

Define 1P = P, and let nP = (n-1)P + P. Then, for example, 5P = P + P + P + P + P (this is pretty intuitive).

A point P is of order k if kP = O and for all positive integers j < k, jP # O.

(These two definitions hold for groups in general, incidentally.)

I found necessary and sufficent conditions on the polynomial coefficients a, b, and c for there to be rational points of order 4 on a given elliptic curve.

Rational points of order two is not difficult, and order three is tedious but not hard. So far as I know, order four had not been found before. We do know thanks to Mazur that rational points can only be of orders 1-10 and 12.

But that's for the rationals. The real points of various orders are completely accounted for--they take on all possible orders, and I'm pretty sure there's not much left in terms of the complex points. I'll check on that though.

I'd be glad to share the details once I hear back from the journal about my paper--I want to make sure it's in print before I spread it around online. Hopefully that will be the case in a few weeks. I'd be happy to start a new thread then.

In the meantime, there's a lot of interesting things about elliptic curves that are fun to play with!

P.S. One extra thing about the definition of elliptic curves is that the complex roots of f(x) are required to be distinct.

Tom Ames

The first two sections of this are a pretty cool introduction to elliptical curves.

The remainder discusses their applicability to crytography (and just as I was admiring the abstract uselessness of it all!)

e3pi

Date: Tue Mar 11, 2003 7:04 pm
Subject: You honestly concerned with earth's people?! I'm concerned with pi

http://groups.yahoo.com/group/SeaPeople/message/188


You honestly concerned with earth's people?! I'm concerned with pi


WHAT REALLY CONCERNS me, is not your "sej Rococo Conoco explanations" or even the dieoff wall the world will soon be smashing against, all I want is to advance what I know about the mathematical constants.

Away from the landbound hordes' chaos and mayhem, on an autonoumous solar powered boat - OOSOOM "Out Of Sight Out Of Mind" of YOU and your's travail, I enter TeX using linux emacs.

I know you don't understand a life's fascination of something you perceive as inconsequential as the constants of mathematics, please also know this indifference is what I share in regards to all you strangers.

"My foothold is tenoned and mortised in blue stone, I laugh at what you call ~dissolution~ and I know the amplitude of time."

RichardMorey

Pi's last message seemed to make a little sense. Maybe they are not Markov chains after all.

Weird....

Muad'Dib

There's nothing inconsequential about mathematical constants. What I find prosaic and frankly a little boring is the apparent need to apply them to all sorts of seemingly unrelated things, when it's much more fulfilling to study them in their own realm for their own sake.

What's wrong with being concerned with both? Surely you realize that every class you've attended, every textbook you've read, was taught and written by people who not only loved mathematics but cared enough about other people to try to communicate what they knew. You wouldn't know what you know without people who care about others.