If you’re a regular reader, then you noticed that no posts went out last weekend — neither a happenings post nor a technical post.
Well, an old friend was visiting for a few days. If you can believe it, I spent time with my friend instead of doing mathematics. (Gee, sometimes I have a life!)
Now I’m back to mathematics — and I have been since he left, when I’ve had the time.
You might also have noticed that the latest technical post was about orbital mechanics. I said it might be. I said that it would be my fallback if both the planned logic post and the first alternative, a color post about my monitor, did not look doable in time.
They didn’t, so I fell back on something I understood. Boy, I barely got that orbital mechanics post done in time. Although I knew the mathematics cold, I spent a lot of time looking at ways to present vectors versus scalars. And that was before I started actually putting dots and arrows on things – it’s perfectly straightforward, but time consuming.
The post makes it clear that I chose to put arrows, and sometimes a dot or two, over vectors, and to increase the display-size of the equations. But before I got to that point, I looked at the results of using boldface for vectors, and rejected it.
I suspect that this weekend’s technical post will be an example to illustrate all the theory in that first orbital mechanics post.
So what’s the holdup with logic, and the holdup with color?
The next post on logic pretty much needs to talk about universal generalization, etc. (i.e. universal instantiation, existential generalization, and existential instantiation). I am struggling between saying too little and saying too much… but I need them in order to prove Aristotle’s syllogisms… so I’ll keep working on what to say.
As for color, I just need to sort out some computations I did at least a month ago. The point is that I now understand something of the nonlinear nature of my computer monitor, and I’d like to share it. This was one of the last two things I once thought I would do for color.
I have, fortunately or unfortunately, found more things to do for color. As I have mentioned, Fairchild’s “Color Appearance Models” looks essential for understanding color perception versus tri-stimulus values XYZ. In addition, Kang’s “Computational Color Technology” has some fascinating material related to Cohen’s methodology. These will lead to more color posts.
There are, not surprisingly, other things going on in my mathemaical life.
In projectiles, I have a fascinating conundrum. The situation is: given a fort up on a cliff, and a ship approaching it… they have cannon with this same muzzle velocity… but there will be a horizontal region measured from the base of the cliff when the fort can hit the ship, but the ship cannot yet hit the fort.
(If the ship is far enough away from the fort, neither can hit the other; if the ship is close enough to hit the fort, it can be hit back. But there’s an intermediate range where the fort can hit the ship but the ship cannot return fire.)
The first part of the question, as you might expect, is: what is the length of that region? In what space is the ship vulnerable while the fort is not?
Simple enough in principle… a little annoying in practice, mainly because there are little niggly details — are there any other kind? — to get right.
The conundrum is part two of the question: what is the length of that region as the muzzle velocity goes to infinity?
Algebraically, I get twice the height — but when I plug in numbers, I get something close to the height. Twice the height is a limiting operation, so I don’t expect to get exactly twice the height when I plug in numbers — but I’m off by a factor of two… even though I would swear I’m using the same solution for both the symbolic limit and the numerical answer.
This is weird. But I’m sure it will work out.
I’m also working on an example for the FFT (Fast Fourier Transform). Once again, the little niggling details have slowed me down — and I think I’ve got one more correction to make — but I got a surprising answer along the way. (Good, that correction will be one of the first things I look at after I publish this post.)
What I’m doing is explicitly fitting trig functions to sampled data. I am absolutely certain — you know, it just occurred to me that that last correction seems like it might very well fix my surprising answer. I’ll try to remember to tell you about this when I put out the post.
Anyway, I know perfectly well that I can use regression (ordinary least squares) to do that fit — as opposed to writing out an explicit solution to this particular problem.
And the coefficients of that fit should be the values of the FFT of the original data.
Yes, there are phase issues, and the representation using complex exponentials versus the representation using sines and cosines. Not an issue; I think I understand that part… famous last words, I know. The point is that the FFT is one way of describing the fit of trig functions to the data.
Along the way, I found another marvelous example of multicollinearity — and I had figured out last December before the holidays how to pinpoint the multicollinearity in a data set. I’m looking forward to presenting this, and I’m delighted to have a really nice example.
It’s particularly nice because it didn’t arise from the usual sources of data. Most of my illustrations of multicollinearity will come from regression on collected data — but this one shows a nice relationship to another area of mathematics, the FFT.
Oh, then there is the question I was asked about modular functions. With a friend’s help, I have been able to reproduce the original drawing. Now all I have to figure out is why it’s a disc!
Like so many things, it will turn out to be a simple calculation… but I haven’t done it yet. Good, this is another of the first things I will do after I publish this post.
It turned out that Mathematica® has a defined function (KleinInvariantJ) that is close enough for the drawing. In addition, you can find useful information on the following two pages of Mathworld:
I had thought I might talk about mathematics and my sense of wonder, and about something called “beginner’s mind”… but I think I’ll put it off, letting this post stand on its own.
That way, I can start doing math now.