Complex Numbers (Book)

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Complex Numbers

Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

http://www.ddewey.net/mandelbrot/

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June 19, 2022 at 12:02am

#1033974

Frickin Lasers

Lasers: not just fun cat toys. Who knew?

Choo Choo, Pew Pew: How a New York Railroad Uses Scorching Lasers to Stay on Schedule
Laser trains may sound like an invention of the 2030s, but they’re already a real tool for the Long Island Rail Road.

First of all, whoever wrote that headline needs to be sent to Gitmo. What are you, six years old?

Second, Long Island Rail Road? On schedule??!

Okay, with that out of the way, the fun sciency civil engineeringy part.

New York's Long Island Rail Road (LIRR) has struggled for years with decaying leaves on the line during what its president Phillip Eng calls "low-adhesion season."

Which I gather is what normal people call "winter."

Trains can struggle to accelerate or stop on the rails' slick surfaces, sometimes locking their brakes and squaring off their wheels, which forces them to be removed from service for repairs.

You'd think that wheels made of hardened steel would be pretty resistant to wear. You'd be wrong.

As the busiest commuter railway in the United States, the LIRR can't afford delays or service interruptions...

Again:

...and has adopted a novel solution to its traction problems: Burn the rails clean with powerful lasers.

SCIENCE!

LIRR president Phillip Eng reported that train cancellations through November have fallen 48 percent year-over-year versus 2018, and that frequency of lighter, but less capacious, ridership-reducing "short trains" is down 32 percent. Delays are down too: 90.7 percent of trains were on time in November, an improvement of 3.8 percent over last year, and delays of 15 minutes or more are down 30 percent.

Well, okay. Maybe because of LASERS their schedule isn't as crappy as it used to be. After all, the whole point is to improve. And to burn shit with lasers.

Still, saying something is "down 30 percent" is kind of weaselly. If all your trains were delayed 15+ minutes, being down 30% means "only" 70% of your trains are delayed. Towards the other extreme, if only 10 percent were delayed, now 7 percent are -- not an amount any individual rider would ever notice.

"No railroad wanted to try it because... It's not service-proven. Somebody's got to be the first to do it," remarked Eng, who is so pleased with the laser train's effectiveness that the LIRR is reportedly working on leasing the technology for future use, as well as outfitting a second train with the same system.

That's always been kind of a problem in civil engineering (of which transportation is a subset): there's a lot of resistance to change, so innovation is a slow process, unlike in, say, mechanical or nuclear engineering. Not just from the engineers (not to be confused with railroad engineers), but from regulatory entities and the general public.

Take, for example, roundabouts -- which are kind of the polar opposite of laser-shootin' trains because they're actually lower-tech than what they replace. Roundabouts are objectively better than stoplights in almost every way, and they're even an improvement over stop signs.

When I was actively designing intersections and shit, a minimal traffic light installation cost about 100 grand, on top of the actual road construction if applicable. Likely it's double that now, and even more if at least one of the roads is more than 2 lanes. Traffic lights can be designed for efficiency, and the switch to LED lamps certainly made a big difference in maintenance costs, but there are still maintenance costs as well as energy required to run the things. Then you have the road sensors in most cases, also requiring maintenance. On top of that, at a stoplight, at least one direction (usually two) is always stopped, creating fuel-wasting idle time and driver frustration. And don't get me started on what happens when there's two lanes of traffic backed up for a quarter mile at a stoplight and suddenly a wild firetruck appears behind them, sirens blaring. People try to get out of the way, but it's chaos.

Compare that to a well-designed roundabout, where traffic rarely stops, and when it does, it's not for some arbitrary length of time. The installation cost is lower than that of a stoplight (depending on how big a statue of some dead white guy on a horse you want to put in the roundabout -- I had to design one in engineering school and I actually labeled it as such). It takes basically no energy to keep running, and as for maintenance, well, you have to maintain the intersection anyway, occasionally repaving it and whatnot. There's more signage, sure, and they tend to take up more real estate, but in general I think they have a cheaper lifetime cost.

But there's serious resistance to roundabouts (also known as traffic circles) in the US, partly because some of the older existing ones, like in DC, suck; and partly because people are just not used to it so they freak the fuck out.

So, anyway. The points are: 1) don't be afraid to innovate and 2) lasers are cool.

Just don't tell me about the swarms of cats chasing the train. I don't want to know.

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