Author Archives: Xander

About Xander

Xander is interested in pure mathematics and mathematics education. He holds a bachelor's degree in mathematics with an emphasis in statistics from the University of Nevada, Reno, and has completed a teacher preparation program at the same institution. After completing his student teaching, he was unable to find gainful employment as a teacher, and has therefore retreated behind the ivory walls of academia in order to complete a master's degree in mathematics. As part of a teaching assistantship designed to help defray the cost of his education, Xander is also teaching remedial algebra at the university.

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

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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 … Continue reading

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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. … Continue reading

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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) … 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

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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 … Continue reading

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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 … Continue reading

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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

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