Category "Math"

Circular law states the eigenvalues of a matrix with random entries of mean 0 and variance 1/n are approximately uniformly distributed in the unit disk of the complex plane. To see this, create a random matrix:

n = 1000
x = randomu(seed, n, n) - 0.5
x *= sqrt(12.0 / n)

Find the eigenvalues:

eigenvalues = la_eigenproblem(x, eigenvectors=eigenvectors)

Plot them:

plot, real_part(eigenvalues), imaginary(eigenvalues), $
      psym=3, $
      xstyle=1, xrange=[-1.5, 1.5], $
      ystyle=1, yrange=[-1.5, 1.5]

This gives a plot like below:

Via the excellent John D. Cook blog. I recommend reading his site if you are interested in a combination of mathematics and Python.

This is fascinating:

People use imprecise words to describe the chance of events all the time — “It’s likely to rain,” or “There’s a real possibility they’ll launch before us,” or “It’s doubtful the nurses will strike.” Not only are such probabilistic terms subjective, but they also can have widely different interpretations. One person’s “pretty likely” is another’s “far from certain.” Our research shows just how broad these gaps in understanding can be and the types of problems that can flow from these differences in interpretation.

For example, below are probability distributions for some common phrases:

via FlowingData

Mathpix is a great idea, executed well:

The Mathpix desktop app allows users to take screenshots of math equations and paste the extracted Latex, all with a single keyboard shortcut.

For example, I wrote this on a piece of paper and took a picture of it:

And Mathpix put the following text on my clipboard:

\sum _ { i = 0} ^ { n } i = \frac { n ( n + 1) } { 2}

Which is exactly right:

$$\sum _ { i = 0} ^ { n } i = \frac { n ( n + 1) } { 2}$$

Mathpix can grab anything that is displayed on your screen.

The Mac app is free on the Mac App Store and there is a corresponding iOS app which is free with in-app purchase.