Gracious Living

Measures and Non-Measurable Sets
November 23, 2010, 14:22
Filed under: Analysis, Math | Tags: , , , ,

This is very late, but don’t worry, I’ll get another one up tonight.

One of the big lessons learned from the Banach-Tarski paradox is that even in something as simple as a unit ball, we can find sets of impossible or undefinable volume. In the discussion preceding the proof, I also mentioned the paradoxes surrounding length of fractal curves in $\mathbb{R}^2$ and area in $\mathbb{R}^3$. Together, these present us with a crisis: how can we characterize length, area, and volume? The answer to this crisis was developed around the turn of the century by heroes like Borel and Lebesgue, and it’s called measure theory.

Banach-Tarski part 2
November 21, 2010, 23:30
Filed under: Algebra, Math | Tags: , , , , ,

So I sort of left you hanging last time.  We talked about equidecomposability, showed that $F_2$ was paradoxical under its own action on itself, and embedded $F_2$ into $SO(3)$.  From here, it just becomes a matter of putting all the steps together: first the sphere, then the ball minus its center, then the whole ball.

Banach-Tarski part 1
November 21, 2010, 06:33
Filed under: Algebra, Math | Tags: , , , , , ,

Okay, here’s the moment you’ve been waiting for: the proof of the Banach-Tarski Paradox.  Here’s what the paradox says:

Theorem (Banach-Tarski).  There are a finite number of disjoint subsets of $\mathbb{R}^3$ whose union is the unit ball, and such that we can apply an isometry to each of them and wind up with disjoint sets whose union is a pair of unit balls.

Or “we can cut a unit ball up into a finite number of pieces, rearrange them, and put them back together to make two balls.”

Products of Groups
November 19, 2010, 22:49
Filed under: Algebra, Math | Tags: , , , ,

Ugh, so, I’ve been really busy today and haven’t had the time to do a Banach-Tarski post.  Since I really do want to see MaBloWriMo to the end, I’m going to take a break from the main exposition and quickly introduce something useful.  There are a couple major ways of combining two groups into one.  The most important one, called the direct product, is analogous to the product of topological spaces.  I know this is sort of a wussy post — sorry.

Isometries of Euclidean Space
November 18, 2010, 21:06
Filed under: Algebra, Math | Tags: , , , , ,

Finite-dimensional vector spaces $\mathbb{R}^n$ come packed with something extra: an inner product.  An inner product is a map that multiplies two vectors and gives you a scalar.  It’s usually written with a dot, or with angle brackets.  For real vector spaces, we define it to be a map $V\times V\rightarrow\mathbb{R}$ with the following properties:

• Symmetry: $\langle x,y\rangle=\langle y,x\rangle$
• Bilinearity: $\langle ax+bx^\prime,y\rangle=a\langle x,y\rangle+b\langle x^\prime,y\rangle$, where $a,b$ are scalars and $x^\prime$ is another vector, and the same for the second coordinate
• Positive-definiteness: $\langle x,x\rangle\ge 0$, and it is only equal to $0$ when $x=0$.

(I’m going to stop using boldface for vectors, since it’s usually clear what’s a vector and what’s not.)  One of the uses of an inner product is to define the length of a vector: just set $\|x\|=\sqrt{\langle x,x\rangle}$.  This is only $0$ if $x$ is, and otherwise it’s always real and positive because the inner product is positive definite.  Another use is to define the angle between two nonzero vectors: set $\langle \cos\theta=\frac{\langle x,y\rangle}{\|x\|\|y\|}$.  In particular, $\langle \theta$ is right iff $\langle x,y\rangle=0$.  In this case, we say $x$ and $y$ are orthogonal.

In Euclidean space, the inner product is the dot product: $\langle (x_1,x_2,\dotsc,x_n),(y_1,y_2,\dotsc,y_n)=x_1y_1+x_2y_2+\dotsb+x_ny_n$.  This is primarily what we’re concerned with today, so we’ll return to abstract inner products another day.

Group Actions
November 16, 2010, 15:00
Filed under: Algebra, Math | Tags: , , , , ,

Today we’re going to take the abstract group machinery we’ve been building up, and unleash it on some sets. When mathematicians say that “groups describe symmetry,” this is exactly what they’re talking about. Say we have a set, and some “symmetries” of that set. Pretty much any definition of “symmetry” will take it to be a bijection on the set, and we moreover expect to be able to undo symmetries, to compose them (associatively), and to use the identity map as a symmetry. These heuristics are just informal versions of the group axioms! The element we’ve been leaving out so far, though, is the set itself on which the symmetries are founded. We say that this symmetry group acts on this set. Below, let’s make this formal.

Free Groups, Generators, and Relations
November 15, 2010, 07:03
Filed under: Algebra, Math | Tags: , , ,

Last time we talked about the cosets of a subgroup. We showed that for a normal subgroup, the left and right cosets coincide, and thus you could multiply any two cosets to get a third coset, defining a group operation on the set of equivalence classes of cosets, or quotient group. We proved the powerful First Isomorphism Theorem, which characterized normal subgroups as the kernels of homomorphisms and showed that the quotient of a homomorphism’s domain with its kernel was isomorphic to its image.

Today, we’re going to introduce a new “type” of group and use the First Isomorphism Theorem to develop a new way of representing groups.