Happenings – 2011 Apr 30

I’ve been all over the place last weekend and during the week, doing a little bit of a lot of things.

I’ve got some things that seem to be finished, in principle if not in practice.

We’ll be done with elliptical orbits as soon as I put out a derivation of Kepler’s equation…. And the second post on projectile motion is so close to completed that it’s silly not to have put it out by now.

I think I’m done with multicollinearity mathematically, but I have to put out two more posts about the Hald data; and I think I want to quickly illustrate multicollinearity in two or three other data sets. A couple of theoretical posts after that, and then a summary and a bibliography, and I should be done with regression for this time around.
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Elliptical orbits: given time, find position

For a change, I decided to split two problems into two posts.

We have found time from position; now let’s reverse that and find position from time. The following problem is Example 2.1 on p. 31 of Prussing & Conway, “Orbital Mechanics” (see my bibliography page).

“An Earth satellite orbit has a semimajor axis a = 4R, and a perigee radius 1.5R, where R is Earth radius. Find the true anomaly at t=4 hours after perigee passage.”
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Happenings – 2011 Apr 23

Well, it’s been a relatively uneventful week.

As I expected… while looking for something else entirely, I moved a small box of papers, and lo and behold – there was my book on differential games. It had never made it off that desk, and had gotten covered up by stuff.

I own a book on self hypnosis… I bought it to try improving my memory… but I misplaced it once, and couldn’t remember where it was.

It was on my desk under a pile of papers. (I have friends who often remind me of that memory failure.)

Some things never change. I still lose books under piles of paper.
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Elliptical orbits: given position, find time

Two of the previous posts about orbits are particularly relevant to this one. Canonical units are described at the end of this post; the geometry of elliptical orbits is described in this post.

I propose to show you how to use “Kepler’s Equation” before I derive it. As we use it, we will see some of the things that enter into the derivation; just as importantly for some of us, we will see why the equation is useful. The equation actually has two parts… it is usually written

M = E – e Sin[E],

where M is called the mean anomaly, but that’s not much use unless we know M:

M = n t,

where t is the time since periapse passage (i.e. since the last time the object was at periapse), and

n = \sqrt{\mu/a^3}\ .

okay… but what is that? Well, the period of an elliptical orbit is

T = 2\pi \sqrt{a^3/\mu}\ ,


n = 2 \pi / T\ .

In other words, n is the average angular speed in radians per second, averaged over one orbit (or any number of complete orbits).
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Happenings – 2011 Apr 16

The big news for the week was that I came down with a cold Saturday night, and that wiped out the rest of the weekend, any possibility of a Monday evening post, and a fair bit of the week. Still, it left me time to think, if not to do.

I think I’m ready to start computing Fourier stuff for “L1 magic”, AKA compressed sensing or sparse representations.

I also think I’m ready to pick up multicollinearity again – I just want/need to write a subroutine to better present the collection of numbers used for assessing multicollinearity. How smoothly that goes will have significant impact on whether or not a multicollinearity post goes out Monday.

My alter ego the grad student made a little progress in the mechanics of beams before my productivity went to hell last weekend. And shortly after I decided it was a shame I only had one reference on statically indeterminate problems (i.e. problems which require the computation of change-of-length for their solution)… I discovered an old battered Schaum’s Outline. I have no idea who gave it to me.

Yesterday I was able to put some time into vector and matrix norms… but I was rather confused by a theorem. I woke up thinking I know what the issue is, but I haven’t confirmed it. Anyway, since I’ve talked about the L1 norm for compressed sensing, and we’ve talked about norms for matrices, I should probably write up something…. I’ve been meaning to, and now it looks like an appropriate topic.

My alter ego the kid is looking for my book titled (I think) “Differential Games”. I just don’t know where it is… apparently not on the shelves with dynamics, math modeling, operations research, or finite math…. Historically, I won’t find it until I go looking for a different book.

And that’s it for now.

On to math.

Happenings – 2011 Apr 9

Let’s just pick this up where I left off last Saturday.

The list of things I do not understand continues to grow faster than the shorter list of things I do understand… but at least the shorter list is growing too.

The two mechanics problems – of course – worked out. I still don’t understand how things went wrong, but I know what went wrong.

In the case where Mathematica® was hung up, my second equation was no longer an equation… it had become “True”. In the case where my intermediate equation appeared to be wrong but the answer appeared to be right… for reasons I do not understand, Mathematica ended up with Ra/2 where I expected Ra (the reaction force at point A). In other words, its variable Ra was twice the reaction force – but the reaction force was still correct.
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backward selection and stepwise: code

Just about a month ago, in a comment, I posted my code for backward selection. I provided virtually no description of the code. I also said that I was too embarrassed to post my code for forward selection.

Well, I decided to do something about that – I have rewritten and tested my code for forward selection. It is, of course, still called “stepwise”, because the name is too deeply ingrained in my head.

While I’m showing you my new code for stepwise, I might as well discuss the backward selection code, too.

In other words, this post is about Mathematica code rather than about mathematics. Still, you might be able to adapt this to another programming language. In addition, it will show you that I have made at least one relatively severe assumption.
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Happenings – 2011 Apr 2

For some reason, I’m having a hard time getting going today. Well, let me just put one word in front of another and see how it goes.

Let me start with compressed sensing – because last Monday’s post has been very popular. I have, of course, ordered two books. More interestingly, I think, there are two more problems I want to work out… a sound signal composed of sinusoids and spikes… and a 2-D image.

I think I’ve almost sorted out in my head what I have to do for such a sound signal… but that means I haven’t even started this problem.

I often think that I am the kind of person who plunges directly into computation before he figures out what he really needs to do. Well, that’s not quite true – in the case of this sound signal, I’m still trying to figure out just what I need to know about it. On the other hand, once I have an idea of something I can compute, I’ll try it out… and learn as I go.
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