## Carnival of Mathematics #56

Welcome to the 56th installment of the Carnival of Mathematics!

Qiaochu Yuan writes about Kraft’s inequality for prefix codes and the probabilistic method, and also on linear relationships between square roots.

Prolific blogger John Cook writes about an intriguing nonlinear differential equation $\dot{y}(t) = t^2 + y^2(t)$, with initial value $y(0) = 1$. Suppose we would like to compute $y(1)$. Numerical methods produce rather bizarre results. The reason is that there is no solution at $t=1$ (finite escape time, anyone?). The moral of the story is that the theorems on existence and uniqueness of solutions to initial value problems do indeed matter.

Rick Regan discovered that positive integers of the form $10^n -1$ have binary representations where the $n$ least significant bits are exactly $1$. For example, $999_{10} = 1111100111_2$. It’s interesting to note that this was a somewhat serendipitous finding.

Harrison Brown has a post on Roth’s proof of Roth’s theorem on arithmetic progressions from a combinatorial viewpoint.

Dave Richeson writes about the prime $8675309$, twin primes and Pythagorean triples.

Américo Tavares lists three gamma function identities.

Mike Croucher writes about Philipp Kühl and Daniel Kirsch’s awesome Detexify project, a $\LaTeX$ symbol classifier. This is not mathematics per se, but it is an interesting tool for those who need to type mathematical expressions. And if you are interested in machine learning, you will definitely be intrigued.

Since there were few submissions to this Carnival, I take the liberty of including some more links of interest to me:

• Math Alive was a course on mathematical concepts that have changed the world. It covers many interesting topics: Cryptography, Error Correction Codes, Data Compression, Probability and Statistics, Chaos and Complex Systems, Graph Theory, Voting and Social Choice Theory. Take a look at the lecture notes. The course was taught at Princeton University by Ingrid Daubechies.
• Edsger Dijkstra once said that “Computer Science is no more about computers than Astronomy is about telescopes”. Computer Science Unplugged is a very interesting initiative whose aim is to teach the principles of Computer Science via a series of recreational learning activities that are suitable for kids of all ages.
• A very interesting applied problem I recently read about is the German tank problem. During WW2, the allies wanted to estimate the number of tanks the Germans were producing each year, so that they could assess the likelihood of success of an invasion of Continental Europe. Intelligence reports were not conclusive. The allies’ statisticians were able to estimate with great precision the number of German tanks being produced from the serial numbers of captured German tanks. The fact that the Germans, in typical German fashion, numbered their tanks sequentially turned out to be quite a blunder ;-)