More Music

I spent the afternoon cursing deer, and the evening chatting with Laura.  Here's a favorite of mine for you.

Two AM and I'm awake? Seriously?

Insomnia really sucks.  I didn't deal with this in the past when I led a lifestyle that involved a lot of exercise, but now I'm in suckville.  Perhaps it's time to reconsider lifestyles.

On another note, I've been debating probability again tonight. (Yeah, I know, I should go to bed.)  Probability is so weird.  Like this: say I flip a fair coin ten times, and it comes up HTHHTTHHTT.  What are the odds of that sequence?  Exactly the same as getting heads ten times in a row.  About one out of 1024, to be exact.

Am I branded with the mark of the geek that I know all the powers of 2 up to 16?  And does that age me?

Two, four, eight, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, ....

My research advisor of old would be proud to see that I wrote all the numbers under "10" in english, I'm sure.  She sure docked me on that point a lot, back in the day.  But bless her for her it.  To this day I still get compliments on my proposal documents for clarity and style, thanks to her.  Although I'm not certain that it works in this context.

Sigh.

On Chaos

Postulate a model of a population of creatures that live a year,  a model that could be used to predict what the population of those creatures will be next year, given this year's population and some lumped parameter that captures the combination of their ability to reproduce and the predatory influence upon them. One such model is:

x_n+1 = rx_n(1-x_n)

Here, x_n is the population at year n (ranging between 0, or no individuals, to 1, or the maximum number that the habitat can support), and r is that lumped parameter.  This particular model has been used for (among other things) predicting populations of temperate latitude insects such as univoltine lepidoptera,
whose adults emerge in the spring, mate, lay their eggs, and die. The eggs in their turn hatch into caterpillars that feed during the summer and overwinter as pupae. Come the following spring, the cycle repeats.

This type of equation is called a map.  This one in particular is called the logistic map.  What this map basically says is: next year's population is proportional to three things, (1) the reproduction/predation constant for this year, (2) the current population, and (3) the remaining empty carrying capacity of the habitat to support new individuals.  (Here a low value for r means that predation is high and reproduction low.)

Here's what the map looks like, when plotted in cartesian coordinates:

Simple, eh?  A concave-down parabola.  The maximum is at r/4, by the way.  This is all pretty simple stuff -- basic first year algebra, the kind of stuff that we took in high school.  But it starts to get interesting if you start putting real numbers into that map.  Say you start with some value, x₁, and you crank through the map and compute x₂ (having chosen some arbitrary value for r). Now say you keep going, computing successively more iterations. What happens? Well, maybe this:

A bit of explanation. You see on the right the concave-down parabola, but there's also a diagonal line.  This line is the line x_n+1 = x_n.  That is, it's where the x-axis equals the y-axis: 1=1, 2=2, etc.  The reason it's there is because this is a map: we choose an x₁, which maps to a point on the parabola, and that point becomes x₂, which maps to a new point on the parabola, ad infinitum.  So the diagram on the right is showing this successive series of steps, in what is called a Cobweb Diagram.  (The graph on the left is just illustrating the yearly change in population, how much x_n+1 differs from x_n.)

Why do we care about this?  Well, if you look carefully, you can see that iterating this map with these values is causing it to collapse to a single point -- the very point where the diagonal line intercepts the parabola.  If you enter that value into the map, you'll get the same value back out.  In mathematical terms, this is called a stable attractor, and in biological terms, it means that the population has reached homeostatis -- predators and environment and reproduction are at a point of equalization, with the population regenerating itself each year (neither growing nor declining).

Now, here's the thing: if there are stable attractors, then there are also unstable attractors.  Populations that grow and dwindle regularly.  A boom, a bust, a boom, a bust, etc.

Nonlinear dynamicists such as myself call that sort of behavior a 2-cycle: there is a single attractor, with two orbits around it.  Boom, bust, boom, bust.  But there is nothing magical about the number two; the Logistic Map can exhibit 4-cycles, too.  And 8-cycles.  And 3-cycles: it doesn't have to be even.  But here's the thing: as you change "r," the cycle changes happen regularly.  First there's a single orbit.  Then two.  Then four. Then eight.  In fact, it's so regular that you can even plot it the number of orbits as a function of "r," and it looks like this:

If you've read this far, then at this point, if I were you, I would be saying WHAT. THE. FUCK.  Because, seriously, what the hell is that diagram, and how did it come from the simplest possible conic section formula known to mankind?  As in, a parabola?  The simplest thing that we learn in algebra?  In HIGH SCHOOL?

Brief explanation: at approximately r=3.0, the Logistic Map enters a 2-cycle.  Hence the branch there, and the two paths.  At about r=3.45, it enters a 4-cycle.  And so on.  The interesting bits are the busy dark areas: that is Chaos.

In the simplest, we find the most complex.

Focus on what is important

Zooey is our calico.  Or, should I say, Laura's calico.  But I've kind of claimed her as mine, too, since we have become boon companions.  She tours the garden with me when I go outside, she sleeps on a cushion in my office when I work, and she grumbles and growls at me when I don't pay her enough attention.

Tonight, I just finished practicing some songs, and she came in the room to say this:

After she made herself known like this and was satisfied that she was the center of attention, she proceeded to deal with pressing cleanliness matters:

That being done, she wrote in that all was cool and I was free to do my thing, and that she would stretch herself out as a compliment to whatever the hell useless work it was that I was doing:

What really matters?

Nothing to see here, move along

I took today off, which combined with the US holiday on monday gives me a four day weekend. Yup.

What to do on a gloriously sunny day off in the pacific northwest? Well, I met up with my buddy Jaime and we cycled down the Interurban Trail to Larrabee State Park. It's pretty great, and long enough so that the casual walker is left behind.  Came home, watered the garden, ate dinner, watched Friday Night Lights and and episode of The Wire with Laura, and now I'm winding down for bed.

Tomorrow: U-pick strawberries (nearly the end of the season!), offload a financial monkey from my back, and then get stuff from Lowe's to deal with my rapidly deteriorating garden paths.  I have beans and lettuces growing like mad, and even one small tomato!  I know, everyone else is harvesting tomatoes now.  But this is the pacific northwest.  And I have eight sunflowers growing strong, despite the slugs.

Blueberry season starts soon, and I don't even bother with raspberries.  There are just too many of them.  But then comes blackberry season.... NOM. NOM. NOM.  Wish that I could grow peppers.

Oh FFS!

When I lived in Germany, I paid $80/year for health insurance. It sucked, sure: when I got sick, I had to wait in line at a local clinic starting at, like, 6:30am, in order to make sure that I was seen by a doctor before they closed at 1:00pm. But then, Germany's health care system is a real piece of work -- in the crap sense.

But now I'm back in the States, and for fuck's sake, can't we just have single payer already? Come on, Canadians, tell me it's worth it. You folks in BC: what do you think? I'm paying $300/mo now for the privilege of dealing with broken websites, piles of paperwork, and basically not getting my job (which I love) done.

All I know is that I've spent AN ENTIRE WORK DAY trying to weed through paperwork from my new employer about health plans in order to figure out the right plan, and this is just for me. FSM only knows what hell I'd be in if I had a wife and kids.

Oh, and I get to do this again in two months. Why? Because when you work for a company that hires an independent HR firm to run their payroll, you basically work for that HR firm. And when their fiscal year starts in September, you get to redo all the paperwork in September.

Jesus H. on a raft, sometimes I wonder how americans manage to put their goddamn pants on in the morning.

Wednesday Night Music

More Tarrega tonight.  I actually spent a good amount of time looking for the best version of this tango, and the one I finally found was the first one that I saw.  And it's not bad at all, although the user's name is... questionable:



But I kinda like how the dude is wearing a sport coat.

In other news, I have tomatoes growing, finally, and black beans too. Finn is chasing Zooey up and down the hallway as I write this, and Laura laughed and smiled as she went to bed.