Sunday, July 26, 2009

What is the Nature of Waste?

First, don't take my ruminations in this post to mean that I am going to become a profligate consumer of things because I no longer believe in scarcity.  I've got a natural inclination to be frugal, and that includes use of resources on general principles.

I was looking at a friend's recirculating waterfall.  It is very pretty, but I asked myself if it was a waste of electricity just to entertain us.  The conventional answer would be yes.  But then I realized that Nature does the same thing, using solar energy on the ocean and lakes to lift water up and then drop it to run down the streams, rivers, and waterfalls to the ocean.  Thus I arrived at the central question of this post: Why do we consider certain things to be wasteful when done by man, but not when occurring in nature?

Another example is a mountain spring.  Where I was last week there was a natural spring with good water.  Of course it trickles out of the mountain continuously, and except for the small amount caught when a person is there, it runs down and into a river and down the mountain.  Now if I were to run my pump to bring water out of the aquifer, and left my hose to run continuously into the nearest stream, most people would say I was wasting a tremendous amount of a precious resource.  Again, one example is naturally occurring while the other is performed by man, but what is the difference in the net effect?

We can ask the question not only in cases where man uses a resource, but where he fails to capture it as well.  Since Nature is using energy to lift water which is then running downhill, are we negligent in not capturing every bit of that energy which is being "wasted" in friction with the rocks?  Should every waterfall be diverted and generate hydroelectric energy?  What about all the solar energy which falls all over the earth?  Are we negligent in not harvesting it?  For that matter, we can even ask about all the solar energy which misses the earth in all the other directions.  Is it going to "waste"?

My answer to this question is that it must not be the nature of the event itself which characterizes it as wasteful, but its impact on what are considered limited resources.  Since Nature has plenty of solar energy available, using it inefficiently to keep the water cycle going is not wasteful.  Since we have limited energy available, using it to no net effect is wasteful.  This leads to the conclusion that if we figured out how to harness copious energy, and weren't short of the materials to do so, and figured out how to use it without side effects like CO2 or excess heat, that then there would be nothing inherently bad about using as much as we wanted.

An interesting and relevant corollary of this is that you can't place blame on previous generations for waste of resources they didn't yet know were limited.  The fact is, almost everything we do is wasteful, but just like the cycles in nature, we find them necessary to life.  Every time I go hiking or cycling and return to the starting point, I've wasted energy which must be replenished with food, which has an impact on resources to grow and ship.  Any trip taken just for entertainment is a complete waste outside of our own mental health.  And yet, we find all these things necessary to make life worth living - just as Nature must use Her resources to support life in general.

So, my conclusion?  Do try to be smart in your use of resources, so we don't run out.  But don't feel guilty about using a reasonable amount to make your life worth living.  After all, if there is any point to our existence at all, it is to enjoy the world around us.

Sunday, July 12, 2009

Observing Instead of Judging

This morning I was thinking about something my daughter had done the night before - leaving some food uneaten.  I was reflecting this morning and thought "I don't like food wasted, but at least she knows to stop eating when she's no longer hungry, which is a good habit".  But then I realized, why do I need to decide whether her actions are good or bad at all?  Does every thing in the world that I see or hear about need to be judged by me as to whether it is good or not?  In most cases, my opinion is either irrelevant, or is not needed.

So, whereas the standard advice would be to "try and find the good in everything", from now on I will try and remember to avoid the thoughts "this is good" or "this is bad" - some things just are.  I know this will be a hard habit to break.  As parents we expect to have to evaluate every situation to determine what actions we will take to help "form" our kids.  Since we have stopped trying to form our kids to let them learn on their own, it seems that making the judgment when I am not going to act on it only serves to build my frustration, and negatively affect my internal image of my kids.

Note, for you who aren't radical unschoolers, this does not mean that I am not parenting, and it does not mean that I can't have an opinion on things.  It just mean that if I say something to my kids about something they are doing, I will state the effect it is having on me (It bothers me when...), but I will not tell them that it is "bad", and I will not try and coerce them to change.  I will just hope that they will take that information on the effects of their actions into account later, along with all the other factors that may influence their decision.

Wednesday, July 1, 2009

Rational Approximation to Roots Continued...

This morning I was able to complete the symbolic expression for the limit of the series.  For large x, the expression (1 + a(bx-c))x can be expanded using the binomial theorem to 1 + x ⋅ a(bx-c) + 12 x (x-1) (a(bx-c))2 + 16 x (x-1) (x-2) (a(bx-c))3 + … ; only keeping the highest power of x in both the numerator and the denominator, they cancel and it becomes ∑ 1n! (ab)n

This is where I was last night, but this morning I realized that it could be a MacLaurin series.  A little browsing in the CRC book and sure enough, it is the series for ey, with y=ab.  So, the actual limit is eab, or solving for ab, we want ab = ln N; indeed, ln 2 is around .693, and 9/13 is a good approximation for it.  Now that I know exactly what irrational number I am trying to have for the ratio ab to converge to any given N, I can use continued fractions to choose good a and b to approximate that ratio.

By the way, to get an exact result at x=1, c should be set to b - aN-1.

However, using Excel to plot the actual values as a function of x, it becomes clear that the convergence becomes slow for even N around 10.  (7, 10, 3) gives values within .015 of 2, but the first few terms when converging to 10 are as low as 8!  One can mitigate this somewhat by raising c such that the values for x=1 and x=2 are poor (do we really need an approximation to the 1st root?), bringing the errors for the next terms down, but this only reduces the error by about half.

So, if you happen to need very high roots, like the 200th root of 10, then this is still an interesting technique.  On the other end, for roots of numbers less than 3, all the approximations are very good.  Since this is not limited to integers, should you need to generate roots of, say, 1.2, this technique will give extremely close approximations.

I did realize that the convergence can be greatly improved by raising the order of x in both the numerator and denominator, effectively creating additional terms which only have an effect for small x.  But, since I started this exploration with the observation that there was a sequence which was easy to remember and compute in my head, I decided to stay with that theme and stop here, rather than create much more complicated expressions.  (At this point, were any of the non-math-interested folks still reading, they would go "What, you mean the previous equations weren't complicated?!" :-)