I continued working on linear programming and quaternions last weekend and a little during this week.
As I said last week, I know how to get “the final tableau” of a linear programming problem using only the problem statement (“the initial tableau”) and the optimal solution.
What I’ve been working on is “sensitivity analysis”, which requires the final tableau. Although I know most of the rules, I’m still trying to fully understand exactly why the rules work. I’m also, in the back of my mind, composing an introduction to linear programming problems.
The story with quaternions is different… I thought I was done with them. But then someone posted a question about them on the sci.math newsgroup… and I thought it was an interesting application of all the work I had already done.
Well, it did require that I add a third Euler angle sequence to my code. (No, I did not originally code up all 12 possible Euler angle sequences… just the two most common in my experience. But adding a third was not difficult.)
I hadn’t decided whether or not to post this problem and its solution… until I remembered that I was going to the local Highland Games next weekend, so it would be a good thing to have one easy post to write up this weekend… then maybe I might be able to write a second post for next weekend, despite losing a full day.
(And I have still never written up the second projectiles post, so that is a possibility too.)
I also realized something that might be useful for my own work.
I have trouble switching gears. If I come to a stumbling block in one subject (today that is linear programming: I put in an hour before starting to write this, and I’m already sort of stuck on something), I want to switch to another subject (and today I think that would be either controls or logic or aircraft; but I would prefer to do almost any form of mathematics rather than none, on Saturday and Sunday. And just in case you’re thinking, “Math on Saturday? Get a life!” – this is my idea of living.)
The challenge is that I haven’t looked at controls, logic, or aircraft in at least a week (aircraft), or two (logic) or three (controls)… and it just feels like starting all over again, and sometimes I find it difficult to generate enthusiasm.
Going for a walk around the block often helps, but not always.
What I realized this morning… while talking to my diary about mathematics, which I try to do every morning before I actually begin doing mathematics… was: If I have a problem, a specific problem to pick up, then it is much, much easier for me to switch to a different subject area. It’s the problem that pulls me, after all, not the subject itself in general.
So, instead of just looking at the books in my vicinity as a reminder of what subjects I might pick up, I am deliberately trying to recall if there was a specific problem I was working on the last time I put down each of those books. And I’ll try to note, when I put a book down, what I was working on when I stopped. (Instead of just writing “resume” in the Mathematica® notebook as a bookmark.)
No, that’s not an earthshaking realization. But anything that helps me be more productive at doing mathematics is A Good Thing.
While I’m on the subject of specific problems, let me share an experience from my graduate student days.
I used to put off grading homework for the class I was teaching until late at night. That is, I worked on the classes I was taking… on my own homework… first, before I was beginning to get tired.
One night, after I finished grading homework, I thought to take one last look at one of my own homework problems, one where I hadn’t even understood the question. I read it… shook my head… and just dropped it in the center of my desk.
When I walked into my office the next morning, I picked up that piece of paper – just to move it out of the way – but I read it again while it was in my hand…
And discovered that I understood the question and knew how to solve it.
You had better believe that I made it a deliberate policy from then on, to read exactly one problem carefully before I left my office at the end of the day, and to leave it sitting alone in the center of my desk, where I could not miss it in the morning.
My subconscious is a pretty useful tool. I’d be very surprised if yours wasn’t also.