Category Archives: math

Anything mathematical.

## Linear models: reversing the predictors and the predicted

Consider $n$ observed data points $(x_1,y_1),\dots, (x_n, y_n)$. We think they might satisfy a linear model $y = ax + b$. Finding the coefficients $a$ and $b$ is called linear regression, and the most typical way to find them is the method of least squares: that is, we find $a$ and $b$ that minimize the […]

## Relax, PhDs: applying to 100+ jobs is normal

Applying for jobs after a PhD and my postdoc was one the weirdest things I ever did. I haven't written too much about it before, but because it is so bewildering I thought I'd give out some stats on how my application process went. The most obvious statistic is the number of jobs to which […]

## A short survey of von Neumann regular rings

I've talked a lot about von Neumann regular rings on this blog, so I thought I'd write an informal short survey on them, collecting some facts we've already seen and many new ones. It should give you an idea of what von Neumann regular rings are. Most of the facts that I did not explicitly […]

## Roger Ming's theorem on von Neumann regular rings

We say that an associative ring $A$ is von Neumann regular if for every $a\in A$ there exists a $x\in A$ such that $axa = a$. That is a rather strange condition, isn't it? But, you can think of $x$ as a pseudoinverse to $a$. This weakening of inverses has a homological counterpart: if every […]

## 50 Awesome facts about prime numbers

A prime is a natural number greater than one whose only factors are one and itself. I find primes pretty cool, so I made a list of 50 facts about primes: The first twenty primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, […]

## Exotic dimensions used in ring theory

Do you ever get the feeling that mathematics uses the word dimension a lot? Well, that's for good reason. The concept of dimension is fundamental in mathematics. What is dimension? You can think of dimension as a numerical invariant characterizing the number of parameters required to do a certain thing. For example, for vector spaces, […]