Tuesday, June 30, 2009

Rational Approximation to Roots

I've been playing with a new math observation. It started when I noticed the following sequence for the nth roots of 2: (1 + 717)2 ≈ 2;
(1 + 727)3 ≈ 2; (1 + 737)4 ≈ 2; (1 + 747)5 ≈ 2. This can be generalized to (1 + 7(10x-3))x = ((10x + 4)(10x-3))x ≈ 2.

So, is this an isolated fluke, or are there expressions like this for all roots? It turns out there are multiple expressions (1 + a(bx-c))x which converge to any N for large x. For 2, the combinations (2, 3, 1), (5, 7, 2), (7, 10, 3) and (9, 13, 4) give increasingly closer convergence to 2. The ratio ab seems to be about .694, but I haven't yet mastered the limit expressions to get a symbolic expression for the number. I've found through trial and error combinations for roots of 3, 4, 5, and 10 - however, the convergence seems slower for larger N, meaning that the approximations are very poor for the lower roots (those you're more likely to use).

So, I haven't yet determined if this discovery is significant or trivial, but if you should happen to be at the sort of party where people are impressed by mental math (I haven't yet found one), you can rattle off that the 7th root of 2 is roughly 1 + 987. Or, more likely, you'll just have a handy small-number ratio for a root if you're doing math by hand.

Wednesday, June 24, 2009

Desire vs. Obligation

We've just returned from the SHINE unschooling conference in Niagara Falls, Ontario.  I had a relevation, but it didn't come at the conference.  It came as we arrived home.

For ten days, we had been touring, seeing museums and natural wonders, and attending the conference, all very enjoyable.  As we neared home, though, my thoughts started to turn to what things would be piled up and need to be done to catch up from being away.  And that's when I realized that part of the reason I'd enjoyed myself so much was that I'd been doing things I wanted to do, whereas in my normal life I almost exclusively do things I feel I ought to do.  I have reached a point where I run my life almost exclusively based on perceived obligations.

I constantly choose to do tasks based on the idea that I will feel better when things have been completed.  I've found in particular I often feel unhappy if I didn't accomplish anything after work, no matter what other enjoyable activities I've done.  In reality there is a never-ending list of other tasks to be done, so the pleasure is short-lived.  The alternative, of course, is to not delay happiness, but do things which are enjoyable right now.  I'm not sure how I developed such a strong sense of accomplishment as reward, but it seems consistent with Puritan values, so I'm guessing it's not that rare in American society.

As they say, realizing you have a problem is half the battle.  So, what am I doing with my revelation?  Well, I'm trying to simply choose at every opportunity the action which will actually bring my pleasure in the doing, not in the finishing.  There are still things which will have to be done, but I intend to do them at the time that I look at them and think, "I would really be happier to have this done right now", not with the anticipation that I will be happier in the future if I don't have to do it then.

One other component of this change in outlook is to reduce the number of things which are perceived as obligations.  Keep up with email, especially mailing lists?  Don't have to.  Blogs I'm subscribed to?  No obligation to read anyone.  All the filing, cleaning, and reading around the house?  I've been not doing it and feeling guilty for a long time, so if I continue to not do it but lose the guilt, I'm ahead, right?  In return I hope I'll find the time to hike, ride bikes, play with the kids, and go to events where before, I might have felt I needed to stay home and get work done.

Sunday, June 7, 2009

Bullet Trains are Sexy, Basic Math is Not

I've been thinking about the statements by Amtrak's CEO Joseph Boardman and Chairman Tom Carper that the way to go faster by train is to improve the slow sections and delays, and the negative reactions accusing them of having no vision (on Infrastructurist, and Chicago news), and I just can't let it go. The thing is, the Amtrak guys are right. Unfortunately, the best solution is not always the one that will sell politically.

Consider a train journey between two cities 180 miles apart. 160 miles of track are wide open, no congestion, etc. Near the two cities and other congestion points, there are 20 miles of track requiring slow speeds and waiting. The train goes 80 mph when it can, but only averages 10 mph through the congested sections. The trip will take 4 hours.

Option one for reducing travel time is to put in a high speed train. It now goes 160 mph on the open rail, but is subject to the same congestion near the cities. Travel time: 3 hours.

Option two to save the same hour is to improve the average speed in the congested sections from 10 to 20 mph. Which solution is more cost effective? Which one is sexier and easier to get public support for?

I haven't gone and researched the history, but I'm willing to bet that the European countries and Japan already had their trains running without significant congestion and delays before they offered their high-speed trains. When I lived in Switzerland 20 years ago, I don't recall trains having to crawl through sections of track, nor coming to a stop and waiting, aside from a few minutes just outside a station. I've ridden Amtrak and these experiences are typical.

I hope that if we do put in high-speed trains that we at least give them improved, dedicated track all the way into the stations. Otherwise it's just option one above.

I hope it doesn't sound as if I don't want high-speed rail in the US. It would be great. But I think there are some necessary precursors that we need to do first.

Wednesday, June 3, 2009

Where Do Hybrid Vehicles Make the Most Sense?

I like to keep up with the development of various types of hybrid vehicles through sites like autobloggreen. A common observation is that hybrid powertrains will give the biggest benefit to vehicles which stop and accelerate frequently. UPS/FedEx trucks, garbage trucks, and city buses are being developed as prototype heavy hybrid vehicles.

I think there is a group of vehicles where the benefits of a hybrid powertrain should be even greater, though: vehicles which run their engines to power on-board systems while they are stationary. I'm thinking of bucket trucks, various lifts, and a lot of hydraulic construction vehicles. There is an office building under construction near mine at work, and I see (and hear) a lot of equipment with the engine running while it holds a load or a person works on a platform. Autobloggreen recently reported on delivery of a hybrid propane delivery truck which can run the delivery pump from the batteries, eliminating the need to leave the truck running during the transfer.

In many of these vehicles, the accessory system doesn't even need power all the time. For instance, the operator of a bucket truck may not need to move the bucket for minutes, once it is positioned. But, currently the engine runs to keep the hydraulic pump ready. By contrast, batteries don't need to use power just to be at the ready. Note that the batteries don't need to be able to power the on-board systems for very long - the engine can still run whenever the batteries need to be recharged, but it doesn't need to run continuously.

There are many more vehicles which are used in this way. If you are alert to it, you will find work vehicles with running engines frequently. Police are another example. I understand that there are enough computers and radios in a police cruiser that the engine needs to run to power them all while the officer works with them (plus, they want the air conditioning to stay on).

I suspect that the construction and utility industries provide a noticeable portion of our fuel use, and although some equipment will have continuous high power requirements, reducing what we can would be worthwhile.