Happenings – 2010 July 31

As you can see, the projectiles post went out last weekend… well, last Monday evening, when technical posts usually go out. In a way, I’m sorry to have used my reserve post so quickly, and yet that is what it was for: to let me do mathematics almost all weekend.

I picked up quaternions as soon as I finished the diary post last Saturday… and I never put them down again, until I had finished what I set out to do.

And what was that? The newsgroup post I had seen triggered something: I decided that the easy way to find an Euler angle decomposition of a matrix was to use its quaternion representation rather than the matrix itself.

I was right. Of course, I didn’t know that until I had actually worked it out.

I expect that I will be writing a couple of posts about (3D) rotations. First, a rotation can be represented by a matrix, by its angle and axis of rotation, by a quaternion, and by Euler angles. Second, I wanted to be able to move between all four of those representations.

I spent a lot of time last weekend telling Mathematica® exactly what I wanted to do. And, all too aften, discovering that I didn’t really want to do that.

But we’ll start slowly. The first post in this set will be an introductory description of quaternions. Among other things, that will let me keep testing my code, refining it if necessary. (Actually, I already want to change three things.)

The second post will probably discuss rotations per se, and their representation by quaternions.

The third post will show how to move between the four representations.

On the other hand, I haven’t decided what I’ll do this weekend after I get the introductory quaternions post written.

But, as usual, we’ll see how it turns out.

Oh, this blog reached another milestone this morning: WordPress has now caught more than 10,000 spam comments. Good job.

Mechanics: simple projectile motion – 1

introduction

Since I’ve been playing with simple projectile motion recently, I thought I would write it up and post it.

In fact, there will be two posts: the first will not require calculus, but the second post will.

This is far from high-powered mathematics… but I enjoyed it.

We do this modeling in a very simple universe:

  • the Earth is flat;
  • gravity is constant;
  • there is no atmosphere (to be specific, no air resistance);
  • and the earth doesn’t rotate (it’s an inertial system, and the ground isn’t moving while the projectile is in the air).

My coordinate system is the usual xy-plane: x is horizontal, y is vertical, the acceleration of gravity is in the negative y direction.

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Happenings – 2010 July 24

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.
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Logic: proving the valid syllogisms

introduction

Added later: there is an Edit near the very end; I removed some silly text.

In this post I want to “finish off” syllogisms. No, this is certainly not the last word on syllogisms, and is almost certainly not my last word on them… but when I pick them up next, it will probably be to look at them using Boolean algebra.

What I propose to do in this post is

  • prove the four valid syllogisms in figure 1;
  • reduce the number of essentially distinct valid syllogisms from 15 to 8;
  • reduce that number from 8 to 6;
  • prove the two additional valid syllogisms;
  • suggest practical guidelines.

Let me begin by summarizing some of the information we will need to have at our fingertips.
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Happenings – 2010 July 17

My mathematical life continues to be extremely narrowly focused; I’m doing logic.

I hope that the next post will summarize syllogistic reasoning… I think I can pretty clearly see the light at the end of the tunnel for this post. Of course, my favorite version of Murphy’s Law is: the light at the end of the tunnel is the headlight of an oncoming train.

Over lunch I mentioned the following puzzle to a friend, and he asked me to e-mail it to him so he could work on it. This comes from Brown’s “Boolean Reasoning”, p. 132 and pp. 134-135. I really like this puzzle.

1. if Alfred studies, then he gets good grades.
2. if Alfred doesn’t study, then he enjoys college.
3. if Alfred doesn’t receive good grades, then he doesn’t enjoy college.

What’s that all mean?
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Logic: adding and dropping quantifiers

Introduction

In this post, I want to introduce us to the formalism of adding or removing quantifiers.

Let me say up front that what I’m about to do amounts to parking my car at a “vista point” along the road, and looking around. If I accomplish anything in this post, it may be to give us some background and context for taking a course in formal symbolic logic, or for working carefully through a text.

As we have seen before, in three or four of my logic posts, there are two quantifiers, \exists and \forall\ ; they are read “there exists ” and “for all”. The first is called “the existential quantifier” and the second is called “the universal quantifier”. It seems that this material was first devised independently by Gentzen and Jaskowski in 1934, so it is surprisngly recent in logic.

What we want to do is to move comfortably between truth tables and tautologies and rules of inference, which do not contain quantifiers… and the kinds of statements which do contain quantifiers, which we find in mathematics and classical logic.

I will be using material I’ve posted about rules of inference and tautologies in one post, “all” and “some” in a second post, and syllogisms in a third prior post.

We have two quantifiers, and two operations to apply to each, so we should expect to find four rules.

Two of these laws remove quantifiers, and two of them add quantifiers. Their precise statements are covered with barbed wire; let’s begin with relatively vague statements. To put that another way, let’s start by being as vague as people usually are when they use these rules.

“The goal is not to make you a mathematical logician; the goal is to make you comfortable enough with quantifiers.” Exner, p. 68.

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Happenings – 2010 July 10

Okay, no technical post at all went out last weekend. By the time I realized that I was facing mathematical difficulties with the logic post, I didn’t even have enough time to finish off a projectile post.

I did, however, get up very early this morning… and I had been thinking about it during the week… and I think that I have resolved all my difficulties. I’ll see if I can knock off the logic post today… and then I would still have tomorrow in reserve.

The fact remains, nevertheless, that if I encounter more difficulties with the logic today, I will probably switch to another topic for a technical post this weekend, most likely the projectile post. That is, if the logic doesn’t hang together today, I will take a break from it. Enough!

And that’s it. I haven’t thought about any other subject, mathematically, since last weekend. Oh, there are a couple of new questions in the comments, and I have been thinking about those. I’ll try to get to them by the end of the day.