My mathematical life is getting a little more varied. As I get to the point of finally being able to compute with Boolean equations, my sense of urgency abates, and I can look at other topics.
Eliminating a variable from a set of logical equations turns out to be very easy… and this means that I can now solve Lewis Carroll’s “sorities” puzzles.
A sority appears to be a collection of premises in which some two variables each appear only once. (Edit: all the other variables occur exactly twice each.) In principle – and in practice, too, but there’s an easier way – one simply picks two premises with one common variable… treats that common variable as the middle term… and obtains their conclusion. Where we might have had seven premises in… what, 8 variables?… now we have six premises in 7 variables. Keep going.
(I think I’m counting correctly, but the exact numbers aren’t important. What matters is that while the process terminates, it is slow, eliminating two of the original premises while adding a new one.)
In practice, however, it is possible to eliminate all of the repeated variables at once.
In either case, we end up with a statement relating the two non-repeated variables. That’s the answer to the puzzle.
Running through his puzzles is fun, and it was one of the hoped-for results of my studying Boolean algebra, but it isn’t work.
So I went looking for some work to do.
I’m getting ready to put out the first new regression post… an introduction to stepwise regression.
I’m working on the first control theory post… making sense of Bode plots.
In addition, there was an interesting post out on sci.math about quaternions, rotation matrices, and Euler angles. I’m strongly tempted to work on that instead. (I think my kid is complaining that he’s been doing logic all the time, and he wants to play.)
What makes this weekend most challenging, however, is that none of these projects is likely to become an actual post. The regression post is closest… but I need to add a lot of discussion to the perfectly straightforward mathematics. And that’s more difficult. (Well, I think the mathematics is perfectly straightforward… but I learned that mathematics a long time ago.)
The new logic mathematics requires that I organize the background material.
Well, I may end up writing a post about “time on an elliptical orbit”. You may recall that our orbit equations relate position and angle – but they say nothing about when the object is at that position or that angle.
There’s a beautiful geometric derivation… and I could probably write it up fairly quickly. We will need it… we really do want to know where something is as a function of time.
I do not know corresponding geometric derivations for parabolic or hyperbolic orbits… but I have seen analytic derivations for all three. But that should be a separate post.
Oh, I have managed to produce a first post about simple projectile motion. I would like to hold that in reserve, and publish it when I simply can’t get another post ready in time.
So, we’ll see what happens. Regression? Orbits? Quaternions? Projectiles?
Oh, sometime early in the past week, total hits on this blog exceeded 70,000. I may not get very many hits on any one day, but you do keep coming.
Thank you. It’s nice to know you’re here.