Sunday, June 11, 2006

Mathematical Poetry

Finite Simple Group (of order two)

A Klein Four original by Matt Salomone

URL: The Klein Four Group
Movie: Finite Simple Group (of Order Two)

The path of love is never smooth
But mine's continuous for you
You're the upper bound in the chains of my heart
You're my Axiom of Choice, you know it's true

But lately our relation's not so well-defined
And I just can't function without you
I'll prove my proposition and I'm sure you'll find
We're a finite simple group of order two

I'm losing my identity
I'm getting tensor every day
And without loss of generality
I will assume that you feel the same way

Since every time I see you, you just quotient out
The faithful image that I map into
But when we're one-to-one you'll see what I'm about
'Cause we're a finite simple group of order two

Our equivalence was stable,
A principal love bundle sitting deep inside
But then you drove a wedge between our two-forms
Now everything is so complexified

When we first met, we simply connected
My heart was open but too dense
Our system was already directed
To have a finite limit, in some sense

I'm living in the kernel of a rank-one map
From my domain, its image looks so blue,
'Cause all I see are zeros, it's a cruel trap
But we're a finite simple group of order two

I'm not the smoothest operator in my class,
But we're a mirror pair, me and you,
So let's apply forgetful functors to the past
And be a finite simple group, a finite simple group,
Let's be a finite simple group of order two
(Oughter: "Why not three?")

I've proved my proposition now, as you can see,
So let's both be associative and free
And by corollary, this shows you and I to be
Purely inseparable. Q. E. D.

My Poem

by Eve Andersson

There once was a number named pi
Who frequently liked to get high.
All he did every day
Was sit in his room and play
With his imaginary friend named i.

There once was a number named e
Who took way too much LSD.
She thought she was great.
But that fact we must debate;
We know she wasn't greater than 3.

There once was a log named Lynn
Whose life was devoted to sin.
She came from a tree
Whose base was shaped like an e.
She's the most natural log I've seen.


A Young Don From Trinity

There was a young Don from Trinity
Who solved the square root of infinity
While counting the digits
He was seized by the fidgets
Dropped science and took up divinity


A New Solution to an Old Problem

The Topologist's child was quite hyper
'Til she wore a Moebius diaper.
The mess on the inside
Was thus on the outside
And it was easy for someone to wipe her.

By Eleanor Ninestein.

An Analyst, Surname of Nero

An analyst, surname of Nero,
Is my mathematical hero.
Says he, "When in doubt,
I always start out,
'Given $\epsilon$ > 0 ...'."

By James R. Martino,
Department of Mathematics
The Johns Hopkins University
Baltimore, MD 21218.

Thursday, June 01, 2006

Continuous, Nowhere Differentiable Functions

The Weierstrass function $f_a(x) = sum_(k=1)^infty sin(pi k^a x)/{pi k^a}$ (originally defined for a = 2) is an example of a continuous function, but differentiable only on a set of points of measure zero.

The function was published by Weierstrass but, according to lectures and writings by Kronecker and Weierstrass, Riemann seems to have claimed already in 1861 that the function f(x) is not differentiable on a set dense in the reals. However, Ullrich (1997) indicates that there is insufficient evidence to decide whether Riemann actually bothered to give a detailed proof for this claim. du Bois-Reymond (1875) stated without proof that every interval of f contains points at which f does not have a finite derivative, and Hardy (1916) proved that it does not have a finite derivative at any irrational and some of the rational points. Gerver (1970) and Smith (1972) subsequently proved that f has a finite derivative (namely, 1/2) at the set of points x = ${2A+1}/{2B+1}$ where A and B are integers. Gerver (1971) then proved that f is not differentiable at any point of the form ${2A}/{2B+1}$ or ${2A+1}/{2B}$. Together with the result of Hardy that f is not differentiable at any irrational value, this completely solved the problem of the differentiability of f.

Since then, a number of mathematicians have developed examples of nowhere continuous, differentiable (CND) functions and it has been shown that these types of functions are the norm (PDF).

One thing that should be obvious is that these functions are not lines in the geometric sense. These are all limits of other continuous functions, that are mostly differentiable, which converge to a nowhere differentiable function.

In Euclidean geometry, a polygon (or straight line) should only be drawn by pencil and straighthedge. If we were to draw any line, then the straighthedge can be omitted and a person only needs to trace the pencil from the starting point to the finish using whatever path they like. Assuming that the person isn't drunk, the line should be of finite length since we are drawing a line fron start to finish. Thus, any line segment that is a subset of that line, must have an even smaller finite length.

However, with CND functions, if you are to pick a point with a pencil and try to trace it to any other point, you will not go anywhere. The reason being is that the standard definition of a line is not met with the CND function. Since at any point, the function is not differentiable, the line would not know which direction to take. Thus, it would appear to vibrate like an electron held motionless in place.

Therein lies the beauty of such functions. They exist and are continuous, but are not real in the geometric sense.