Review for Week of Apr 24 (and May 1)

We calculated the fractal box dimension for Brownian motion and were unsettled at our rational result of 3/2.

We concluded that W(t + dt) - W(t) / dt = sqrt(dt)/dt and that the limit of that as dt->0 does not exist. Which means that we have a continuous function that is not differentiable anywhere.

We also assessed that this guy is going nowhere really fast.

Moving on to Hausdorff Dimension, we defined it using vague covers of sets Ai of some restricted diameter (delta). The diameter of each (a number) is raised to a power (s) and summed to arrive at a set of #'s, the inf of which is our Hausdorf H^s_d(E).

The limit as delta -> 0 is the Hausdorff s-dimension measure of E, which we proved exists as the lim increases with or without bound.

What I find interesting is that we look for the s value that drops from infinity to zero. It is odd that it changes, and odder that we can use that change as a definition of dimension. It's one of those quirks that make math cool for me.

It is also interesting that it is invariant under all sorts of transformations. That is comforting in a way that few things in this course have been.

It is nice that the zero measure does what you intuitively feel like it should, and it seems like this things follows properties of a group (yes/no?)

This this is ugly to compute and you have to squeeze it. We computed the upper bound pretty easy, but the lower bound was not so easy. We worked up to an ideal cover by designing one that was:

• A_i subset of E
• closed
• interval
• not overlapping at endpoints

Then we were able to take max advantage of the curve and calculus to conclude that the sum evaluates to the length of the curve, then the lowerbound(E) is less/equal the H_delta^1(E).

This means a decrease and folding is possible. I wish I knew more about what that meant. We ran out of time for questions.