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Author Archives: Xander
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 … Continue reading
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 … Continue reading
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. … Continue reading
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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 … Continue reading
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 … Continue reading
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 … Continue reading
Hummingbirds
Grad school is keeping me busy. In the meantime, hummingbirds from our balcony 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 … Continue reading