This post doesn’t feature any real math. You’ve been warned.
Now on to the stuff I found out about. The first two have to do with matlab.
Now, I honestly can’t believe that I’ve been working with matlab for so long and I didn’t know about hold all. Of course, I’ve been using hold on/off a lot. But let me take if from the top.
As a command, hold on informs matlab that we would like to add more graphs on the same plot. It’s useful if you want to have multiple data on the same place but you don’t have all the data immediately or it’s messy to try to plot them all at the same time. Thus you run hold on and voila! All the graphs on the same figure.
There’s only one small problem. Matlab will reset the color order, so you get all of your plots with the same colour.
But if you use hold all, the colour order is not reset. It keeps on changing with each new colour, which I find quite cool. Here’s the result of running the code with hold all.
And yes, the colours follow the same order each time.
The second thing I found about has to do with the plots themselves. Perhaps the observant readers will have noticed the grey part around the figures, not something usual. This is the result of export_fig, a function written by Oliver Woodford. The real difference in quality can’t be seen in the previous figures (blame the jpg compression, always the jpg compression). But you can always check this site to learn more about it.
On the math side of things, I’ve been pretty happy with the courses this semester. All of them are interesting but for some reason, I feel like I’m learning more new stuff on the PDEs class. I was certainly blown away by some of the stuff in that class, like the Cauchy – Kovalevsky theorem, giving conditions for the existence of solutions for a general PDE (somewhat too strict though).
Or the Duhamel’s principle, an idea with which you can get the solution of a linear inhomogeneous PDE, like the Wave equation. It’s a fun principle to think about, cause you can understand all about the why it is true from the simple proof but the “how can this be true” isn’t so obvious.
Whereas the PDEs course is something like a “general knowledge” course, dynamical systems is a crazy train of tools. Tools to check the stability properties of a solution of a system of ODEs near an equilibrium, tools to talk about what’s happening to the solution after a long time, tools, tools, tools. It goes to show that there’s a lot more into ODEs than someone would learn at an undergraduate course (but this of course applies to any subject). It’s also a good example of how many areas of mathematics meet.
Lastly, I’ve been experimenting with the Ant Colony Optimization method, a biomimetic idea for finding the solution of a combinatorial optimization problem, provided that the problem looks (or can look) like a travelling salesman problem. You can find a whole page dedicated to this problem at Georgia Tech. There’s also at least one movie inspired by it.