Pony Break

Mmk time to break up the monotony.  With the Apple family!

Great mixed drink for the winter:  Apple Cider and Cinnamon Schnapps (I tend to use Goldschläger).  Add to taste.

Don't forget the new episode! (taken down)

Applebloom goes on an 8-bit adventure

Stay thirsty, my ponies.

Suspicious Characters, Part 2

It's trace operator time!  For those who don't know already, the trace of a matrix is the sum of the entries on the main diagonal.

Why is it importart?  We won't be proving this fact today, but the trace of a matrix $\rho_g$ is also the sum of its eigenvalues!

Eigenvalues?  Wündebar!  Time to make de magics.

Suspicious Characters, Part 1

For those familiar with linear algebra and group theory...

Suppose we have a group that acts on a vector space.  This situation comes up fairly often; every time we have an object situated in euclidean space, the symmetries of the object are a subgroup of the symmetries of the space itself.  Even if that seems kind of obvious, this is a genuinely useful insight: the symmetries (a.k.a. automorphisms) of $n$-dimensional euclidean space can be represented by matrices, and there are a lot of powerful techniques out there (e.g. determinants, eigenvalues) that only work on matrices.  So studying the matrix representation ought to be a useful technique for understanding the action of the group.

This is more effective than you might realize at first: if both the group and the dimension of our vector space are finite, there is a nice way to decompose the vector space into orthogonal orbit-like-subspaces just by looking at the traces of the matrices.  Over the next few blog posts, we'll be investigating this business with decomposition using trace operators--known as character theory.

With some character theory, Rainbow Dash might have noticed

the different subrepresentations of Mare-Do-Well beforehoof.

Discord in the Axioms

You've all heard about Gödel's Second Incompleteness theorem, right?  If a theory with a finite list of axioms contains the basis for arithmetic, this theory cannot produce statements about its own consistency without contradicting itself.  So we can't ever prove such a theory's consistency; we can only take solace in the notion that we haven't found any contradictions yet.  Kind of a bummer.

Way back in September, the distinguished Ed Nelson, a math professor at Princeton, claimed to have found an inconsistency in Peano Axions--one of the most widely used formalizations of arithmetic--and distributed a sketch of his proof through a math mailing list.  It's hard to get more basic than this: these are axioms that establish the existence of zero, all the common rules for how the equals sign works, your ability to add one to a number that already exists, and (Nelson thought this last one was the culprit) the validity of proof-by-induction.  There might be some question about whether certain bits of modern set theory are inconsistent, but arithmetic?  Nelson was really going for the balls here.

What It's All About

Everypony knows that the pony pokey is what it's all about.  But what about this blog-thingy?

This blog is for sharing the two best things ever: math and ponies.  I hope to crank out a pleasant introduction to some interesting math once or twice a week.  Pony-related updates will occur as needed.  Will I find occasions to combine these two delightful things?

By Celestia's simply-connected flank, I think I should!