Happenings – 2012 Mar 31

The comfortable routine I have settled into has been disturbed… in a good way… I think.

If I can preserve some of the routine today, I may be able to write up another post about trusses. This one will even be a real truss – a Howe roof truss.

Or, it looks like I have the mathematics for reading the parameters of sinusoids from their discrete Fourier transform… and, as a follow-up post, a nice real-world example. Both look fairly easy to finish off.

In the meantime, my alter ego the kid has finished reading – not working! – Oystein Ore’s “Number Theory and Its History”. It is an introductory level book… very smoothly written… available as a Dover paperback.

Although it had some unfamiliar mathematics in it, what I got out of it was that the 8-gallon puzzle goes back to medieval times.

Trusses – Example 2

Let’s try a slightly more complicated truss. You may want to read the previous post, if you have not already.

Happenings – 2012 Mar 24

I briefly considered delaying this post until tomorrow, March 25. On that day, in the year 3019 of the 3rd age, The One Ring fell into Mount Doom and Sauron fell into ruin. Some people celebrate March 25 as “Tolkien Reading Day”.

I think I’ll do mathematics instead, tomorrow… but maybe I should start talking to a friend about watching all of “Lord of the Rings” again in one day… after all, it’s been a year since we last did it.

For a change, I have absolutely no external obligations today… how much mathematics can I get done today? It occurs to me that the major limitation will be cats crying and trying to sleep in my computer chair. The black cat will come here around 5 PM, and I find it hard to resist him. The white cat has been trying to sleep here for the past hour… every time I get up, he jumps into the chair. (There’s no sun this morning, so his usual sunny corner window isn’t.)

As for the past week…

Trusses – Example 1

Given the following picture of three beams… with a given force applied at the apex… let’s see if we can work out the internal forces in the beams, and the reaction forces at the two bottom points.

This is an example of a “statically determinate” problem. We will be able to solve this assuming that the three beams do not bend or compress or stretch.

Happenings – 2012 Mar 17

Ah, I didn’t realize until I entered the title for this post that it was St. Patrick’s Day. And last Wednesday, being 3/14, was Day. I can’t say I celebrated the latter, nor expect to celebrate the former – but who knows? I might remember to wear a green shirt… do I own a green shirt? – when I go out later today.

(Yes, I do sometimes stop doing math and leave my cave on weekends.)

Using the QR Decomposition to orthogonalize data

This is going to be a very short post, illustrating one idea with one example (yes, one, not five).

It turns out that there is another way to have Mathematica® orthogonalize a matrix: it’s called the QR decomposition. The matrix Q will contain the orthogonalized data… and the matrix R will specify the relationship between the original data and the orthogonalized.

That means we do not have to do the laborious computations described in this post. Understand, if we do not care about the relationship between the original data and the orthogonalized data, then I see no advantage in Mathematica to using the QR over using the Orthogonalize command.

Happenings – 2012 Mar 10

Some things are progressing… some are not.

I had thought that I could write up a group theory post on semi-direct products for this Monday… while they are not too difficult to describe, I want some examples to play with… and, sad to say, the examples require more familiarity than I have. On the one hand, this is a good thing: I have identified something else I need to understand in group theory. On the other hand, the post about semi-direct products will have to wait.

I had thought that I could write up a simple post showing how the parameters of a sampled sine curve could be identified from its DFT (discrete Fourier transform). It really is pretty simple – except for one thing, the phase.

I’m sure that once I understand it… actually, by my very definition of understanding… the phase will seem simple. It turns out that I’m very close to figuring it out, I think. (I worked on it this morning, before this post… in fact, before my stream-of-consciousness… and, I haven’t turned the kid loose yet.)

It posed an interesting conundrum. Should I put the post out before I figure out the phase? Or should I wait until I understand that, too?

Well, if I really want this blog to be about the doing of mathematics rather than just the done of mathematics, then it makes sense to post something I only partially understand yet.

On the other hand, it seems silly to post it if a few more minutes will clear something up in my mind.

Group Theory – Building abelian groups

I propose to find all the abelian groups up to order 100. It’s pretty easy, and there’s one nice idea that will simplify things. An abelian group, I hope you recall, is one that is commutative.

(The smallest non-abelian group is D3, the dihedral group of order 6… which is isomorphic to S3, the symmetric group on 3 symbols. All other dihedral groups are non-abelian, and the quaternion group of order 8 is non-abelian. But we’re going to look for abelian groups.)

Happenings – 2012 Mar 3

Many years ago I took a bunch of courses in multimedia studies… among them, 3-D animation. My original goal was simple enough: to learn how to use my computer for more than number crunching.

Of course, that couldn’t have been all that long ago… although the program included a course in the history of multimedia. It was hard to believe, going in, that it had any history. Did you know that Richard Wagner (e.g. The Ring Cycle) was the first dramatist to insist that the lights be turned off for his shows? Composer, oh yes. But impressario, too, and the whole show was important to him. (And that’s about all I remember of the history.)

Ironically, because I was in a certificate program, I was able to get a student version of Mathematica – which put me on the road to using my computer for even more number crunching.