Fractals 6 Slides

On Saturday, I will be presenting some of my recent work, which extends Michel Lapidus’s theory of complex dimensions in \(\mathbb{R}^n\) to a larger class of homogeneous metric measure spaces. In particular, I will be discussing the complex dimensions of subsets of \(p\)-adic sets. In case they should become necessary: slides for Fractals 6.

Abstract: The higher dimensional theory of complex dimensions developed by Lapidus, Radunović, and Žubrinić provides a language for quantifying the oscillatory behaviour of the geometry of subsets of \(\mathbb{R}^n\). In this talk, we will describe how the theory can be extended to metric measure spaces that meet certain homogeneity conditions. We will provide examples from \(p\)-adic spaces and discuss the geometric information that can be recovered from the complex dimensions in these cases.

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UCR Beamer Template

For those that are interested, my UCR themed Beamer template is available here.

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Some Trig

A large portion of the “standard” precalculus curriculum consists of a rather tedious recitation of trigonometric identities. I am largely of the opinion that there are only a couple that one really needs to know—for example, the Pythagorean identity (\(\sin(\theta)^2 + \cos(\theta)^2 = 1\) for all \(\theta\)), one of the angle addition formulæ (e.g. \(\cos(\theta+\varphi) = \cos(\theta)\cos(\varphi) – \sin(\theta)\sin(\varphi)\) for all \(\theta,\varphi\)), and the law of cosines (essentially the Pythagorean theorem with a correction term) immediately come to mind as important results. In the last quarter, while teaching precalculus class, I was introduced to a couple of proofs that I had not seen before. I don’t get the impression that either proof is that unusual or original, but I do think that they are both rather slick. I’m putting them up here for future reference.

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Bread

My wife brought home a loaf bread this weekend. The nutrition label reads as follows:

I’m not entirely sure how one loaf of bread can consist of only 4.5 servings when a single serving is one ninth of a loaf. On the other hand, fractions are hard, I guess. :\

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Seminar Talk in FRG

I gave a talk last week before the Fractal Research Group at UCR. The goal was to introduce my colleagues to the Assouad dimension, and to share some of the more interesting and/or surprising results. My notes (typos and all) are available for those that are interested. During my talk, I managed to get through the sections on homogeneity and the Assouad dimension, including most of the examples involving orthogonal sequences. I did not have time to discuss the Laakso graph. As further reading, I suggest James Robinson’s Dimensions, Embeddings, and Attractors—my notes are shamelessly plagarized from chapter 9 of that volume.

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The Sphere is Weakly Dense in the Ball

Another qual question, this one dealing with Hilbert spaces and the weak topology:

Exercise: Let \(\mathcal{H}\) be an infinite dimensional Hilbert space. Show that the unit sphere \(S := \{x\in\mathcal{H} : \|x\| = 1\}\) is weakly dense in the unit ball \(B := \{x\in\mathcal{H} : \|x\| \le 1\}\).

I find this to be a really surprising and counter-intuitive result. What this exercise asks us to prove is that every point inside of the unit ball in an infinite dimensional Hilbert space is, in some sense, arbitrarily close to the boundary of that ball. This is really striking to me—how can the center of a ball be really close to the boundary of that ball? Because this result is so unexpected, I think it is worth understanding. Indeed, the arguments presented below have helped me to build some better intuition about both infinite dimensional Hilbert spaces and the weak topology.

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Nowhere Differentiable Functions

In an undergraduate analysis class, one of the first results that is generally proved after the definition of differentiability is given is the fact that differentiable functions are continuous. We can justifiably ask if the converse holds—are there examples of functions that are continuous but not differentiable? If such examples exist, how “bad” can they be?

Many examples of continuous functions that are not differentiable spring to mind immediately: the absolute value function is not differentiable at zero; a sawtooth wave is not differentiable anywhere that it changes direction; the Cantor function[1] is an example of a continous function that is not differentiable on an uncountable set, though it does remain differentiable “almost everywhere.”

The goal is to show that there exist functions that are continuous, but that are nowhere differentiable. In fact, what we actually show is that the collection of such functions is, in some sense, quite large and that the set of functions that are differentiable—even if only at a single point—is quite small. First, we need a definition and an important result.

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Notes:

[1]
I’ll (hopefully) talk more about this later.
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An Elementary Duality Result

The following problem is a fairly straightforward exercise that seems to come up fairly often on qualifying exams at UCR, and I think that I finally have a nicely argued proof. I am going to put it up here mostly for my own edification. Please note that the following is intended largely for an audience of advanced undergraduate or graduate students. I will be making no effort to give all of the required background, and assume some knowledge of the theory of Lebesgue integration and \(L^p\)-spaces.

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Hummingbirds

Grad school is keeping me busy. In the meantime, hummingbirds from our balcony feeder:

A hummingbird at our feeder.

A hummingbird at our feeder.

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Fractal Music(?)

In the past, I have treated fractals visually. For instance, in my treatment of the Mandelbrot set, I chose points in the complex plane to represent pixels in an image, applied the iterative function \(z_{n+1} = z_{n}^2 + z_0\) to each of those points until the orbit escaped (or failed to escape after a fixed number of iterations), then colored the pixel based on the number of iterations applied before the orbit escaped. Another visual approach was to compute a large number of orbits and treat their traces like photons hitting a light sensor, thus creating nebulabrot images. I have recently started playing around with an entirely different idea: instead of using the orbits to compute values for pixels in an image, the orbits can be used to generate sound.

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