# A Map of Math?

There has been some recent discussion at Reddit of an attempt I made to turn some data from arXiv.0rg into a visualization of the relationships between research areas in mathematics.  I was never completely pleased with the dataset.

# The Cauchy–Schwarz Inequality IV

The Cauchy-Schwarz inequality,

$\displaystyle\sum_{i=1}^n{a_i^2}\sum_{i=1}^n{b_i^2} \geq \left(\sum_{i=1}^n{a_ib_i}\right)^2,$

holds for any pair of lists of $n$ numbers, ${a_i}$ and ${b_i}$, for $i=1,2,\ldots,n$. I am about to discuss a fourth proof of the Cauchy-Schwarz inequality. I have previously presented a first, second and third proof.  The fourth proof involves some algebraic pyrotechnics where we measure the difference between the expressions on each side of the Cauchy-Schwarz inequality. Continue reading

# Planarity

I recently discovered the game, Planarity. The goal of the game is to take a graph and put it in a form where none of the edges cross. That is to say, to prove that it is planar. If you don’t know what I’m saying; or you think it sounds boring; or you think it sounds complicated; go play it anyway. It’s lots of fun.

# The Cauchy–Schwarz Inequality III

I have previously written about the Cauchy-Schwarz Inequality here and here.  To recap, the Cauchy-Schwarz inequality,

$\displaystyle\sqrt{\sum_{i=1}^n{a_i^2}}\sqrt{\sum_{i=1}^n{b_i^2}} \geq \sum_{i=1}^n{a_ib_i},$

holds for any pair of lists of $n$ numbers, ${a_i}$ and ${b_i}$, for $i=1,2,\ldots,n$.    Now, I have given two proofs so far, and here is a  third.  We will proceed by induction on the number of variables.   Continue reading

# Goolge Wave Invitations

If you have been late to the party with Google Wave, Lifehacker has a donation thread where kindhearted souls with an excess of Google Wave invitations offer them to all who are interested.  You are going to need an active account on Lifehacker to request an invitation from most people, which is not as easy as it might seem.  However, a few people just paste their email in their comment, so you can just email them directly without needing to comment on Lifehacker.  I was able to get an invitation for a friend using Lifehacker’s donation thread so it seems to work.

[Hat tip: Lifehacker]

# The Cauchy–Schwarz Inequality II

I have previously written about the Cauchy-Schwarz Inequality.

$\displaystyle\sqrt{\sum_{i=1}^n{a_i^2}}\sqrt{\sum_{i=1}^n{b_i^2}} \geq \sum_{i=1}^n{a_ib_i}$

Here is another proof of this fun inequality.  We start at a similar point to the previous proof Continue reading