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Category Archives: Mathematics
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
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
LEGO Fractals
My daughter has recently started playing with DUPLO blocks pretty consistently, which has me thinking about LEGO—the toy of my youth (and hopefully her’s in a couple of years, too!). In some ways, LEGO is an ideal medium for exploring … Continue reading