In the interest of not letting too much time go by without a word, a couple of quick updates:

### Thesis: Defended

My thesis defense in April went very well. There is a professor in the mathematics department at UNR who has a tendency to ask really persnickety nit-picky questions with annoying regularity (at defenses, colloquia, and informal conversations). He is rarely wrong in his assumptions, but the questions often distract without really adding much to a presentation. I was quite worried that, during the public portion of the defense, he would come up with a humdinger, but it seems that my presentation was well-prepared and clear enough to avoid stepping on any of his pet peeves.

Following the public defense, I was quizzed by my committee for nearly an hour. My impression had always been that this part of the defense was meant to be somewhat adversarial (in essentially the same way that an exam is adversarial). It turns out that we had a very nice conversation about the mathematics presented, my plans for the future, and monkeys.

I was left with a small list of revisions and was then invited to leave the room while my committee came to a decision. Ten minutes later, all of my paperwork was signed, and I was functionally a master of mathematics.

My Defense Slides and Masters Thesis are available.

### New Ink

In celebration of completing my masters degree, I got a new tattoo:

In my thesis research, asymmetric Cantor sets proved to be interesting objects of study. An asymmetric Cantor set can be described as the attractor of an iterated function system consisting of only two maps. Such systems are highly regular (where “regular” is used here in a somewhat informal sense), thus one might hope that they are well-behaved with respect to certain notions of “well-behaved” that are explored in the thesis. As described in the thesis, we have no such luck, which was, to me, a really surprising result.

The tattoo itself is the first five stages in the construction of a particular asymmetric Cantor set (that obtained by leaving the first half and last third of each interval at each stage of construction). This particular set does not exactly behave badly in the sense of the thesis, but it is an aesthetically pleasing member of the family of sets I studied, thus it seems to be an appropriate symbol of the research that I have been working on for the last two years.

## Thesis Defense

In five days, I will be defending my masters thesis. Whee!

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## MMM LXVII

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## MMM LXVI

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## Finding a Limit

This semester, I am assisting my advisor with his honors calculus class. As he is out of town today, I will be giving the lecture. We decided to show off a rather beautiful little bit of mathematics which makes use of many of the tools that we have developed since September, including an interesting application of Riemann sums. We are going to be trying to evaluate $$\newcommand{\8}{\infty}\lim_{n\to\8} \frac{n}{(n!)^{1/n}}$$.

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For the last several months, I have been working hard on getting my masters thesis written. This is a somewhat tedious activity—the research phase is essentially done, and now I have to get everything typed up all nicely. Other interests have taken something of a backseat. One of the things that I have managed to do in order to stay sane is make a lot of cookies. I have been experimenting with recipes, and came up with something pretty good this week.

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## The Fundamental Theorem of Algebra

### An Analogy

When cooking, we need three things: a collection of ingredients (say butter, sugar, flour, &c.), a description of a technique for combining them (a recipe), and some tools for actually getting the work done (bowls and knives, as well as some level of skill in their use). Mathematics is much the same—in order to practice mathematics, we need ingredients and techniques for combining them. In a very broad sense, the basic ingredients are the axioms, which can be mixed together using first-order logic and an ability to use these to produce theorems.

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## Ten Factorial Seconds?!

Here’s an interesting little problem that came across my desk this afternoon: how much time is $$10!$$ seconds? Is it a duration that is best measured in seconds? days? centuries? And, perhaps more importantly, what is the best way of figuring it out? Think about it for a minute, then check below the fold for the answer, which is a surprisingly round number!

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## GREs

I took the GRE subject exam in mathematics Saturday morning. It was a pretty brutal test. The algebra, topology, and analysis all seemed pretty straight-forward, but the linear algebra and calculus computations kicked my ass (much as I expected). Now we wait 6 weeks to see how bad it really was.

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## Newton Fractals

The question of solvability is one of the driving questions in mathematics. That is, given an equation, can it be solved? and if so, what are the solutions?

There are some equations that are very easily solved. For instance, the linear equation $$ax + b = 0$$ can easily be solved by subtracting $$b$$ from each side then dividing through by $$a$$. That is, $$x=-b/a$$ is a solution to any equation of the form $$ax+b=0$$. Quadratic equations, i.e. those of the form $$ax^2+bx+c=0$$, also have solutions. In fact, they generally have two solutions that can be found using the quadratic formula, and which look like
$\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$
There are also “nice” formulae for solving cubic and quartic (degree three and degree four polynomial) equations. These solutions involve only elementary mathematical operations like addition, multiplication, and taking roots. So what about higher order polynomials? For instance, does the equation
$ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0$
have any solutions, and, if so, what are those solutions?

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