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## Busy-ness! And a love for Maple…

The quarter is off to a running start! I wish I could post more here, but… Whew!  Hopefully this post will make up for my absence!

So here’s a little potpourri for you:

Control Theory: is a wonderful branch of mathematics that takes the idea of dynamic systems one step further by look at how we can use inputs to control the system. Our MTH 610 class is using this book. Right now, the simple idea is (for a certain type of system) that we can model the system with a linear differential equation:

$\dot{x} = A x + B u$ (1)

Here $x$ is a vector function giving the state of the system as is evolves over time and $\dot{x}$ is the time derivative of that function. The matrix $A$ is (if $x \in \mathbb{R}^n$) an $n \times n$ matrix. Now, if we just stopped there, we would have:

$\dot{x} = A x$

which would simply give us a dynamic system. We could watch it move through its states without any outside interference… But, that’s boring! Instead, we want to affect the state of our system, so we need some inputs given by the vector $u$. How many inputs do we need? It depends on the system, but it almost certainly won’t be $n$! So, if we have $m$ inputs, then tobe able to add the vectors, we must multiply $u$ by an $m \times n$ matrix $B$, giving us equation (1).

Well, in actually studying these equations, we end up having to do a lot of linear algebra (as you might guess from the matrices), including finding characteristic polynomials, and matrix inverses. This is where Maple comes in.

Maple: is a symbolic algebra program, and a friend of mine gave me an old copy to use a couple of years ago (and the copy is very old, the system requirements ask only for 8meg of RAM and a 486 processor!). I didn’t really need it for anything, so I never installed it. Well, in working through a control theory problem (Exercise 5.1.14 in the text), I ended up having to fiddle around with several $4 \times 4$ matrices, finding their inverses, and checking my answer by multiplying them all out… I was doing it by hand, and am reminded of Dr. Duchateau (at Colorado State University) who told us that it was in his contract to never do anything with matrices larger than $3 \times 3$ by hand!

Of course, I made errors. Small ones that I didn’t catch, and when I finally got to checking my solution – it didn’t work. So, in frustration I began looking for symbolic algebra systems online. There’s Jacal, which is implemented in Scheme, as well as Fermat – neither of which are very friendly to the novice. At PSU, someone also recommended another bit of open-source software – please chime up in the comments! I forgot who told me and what software you told me about! – and then I remembered the copy of Maple that my friend had given me. The gui version fails to recognize my keyboard for some reason, but the command line version works fine, and after only a couple of hours of learning, I was able to completely finish the problem in minutes! As well as the second problem (also a $4 \times 4$ system)! I caught several mistakes and also was able to check my solution easily.

The computer lab on the fourth floor of Neuberger Hall has mathematical software, but the department website doesn’t tell us what software that might be. I know that $\LaTeX$ is one of the things available, and perhaps Mathematica or Maple (or both?)… I will try to find out and post an update here.

Games: Where did that category come from? Well, it turns out that control theory is used to optimize (as you might guess from the differential equations, we can look for minima/maxima) — specifically to optimize the inputs to get us to a certain desired state $x$ of the system within the shortest time (or using the fewest resources, or whatever your constraint is). In one sense, we can think of this as deciding a strategy to solve a problem – which brings us back to games theory. Saij might have some commentary here (being the Lyceum’s resident games-theorist), and next quarter there will be a class at PSU on differential games which will make the connection explicit.

That’s all for now- time to hit the gym and then over to PSU. I hope to see some of you there!

ex animo-

Felicis