Last week I came home happy one day because of a simple little math expression. The Fibonacci sequence has the cool property that for each term, if a prior term's index is a divisor of the current index, then that term is also a divisor of the current term. So, for example, F15 is divisible by F3 and F5 (and an additional factor). More formally, a divides b implies Fa divides Fb. There are other sequences with this property, including 10n − 1 (or any bn − 1). But the Fibonacci sequence is a linear recurrence relation (each term is the sum of the two previous terms) and the sequence of one less than powers of ten is exponential. I figured there must be some way to characterize what types of sequences can have this property.
Well, I still don't know if there are other types of sequences with this property, but I do now know why these two types share it. I came across this relationship in a paper I found online, and as soon as I saw it I realized it could be generalized. They are actually the same type, and can each be expressed as the other. For instance, 10n − 1 can also be obtained using Fn = 11·Fn-1 − 10·Fn-2, and each linear recurrence sequence can also be made into a difference-of-powers sequence, though the base "b" may not be rational.
As with many techniques in math and physics, once you know there is an equivalence, it is trivial to set up a set of equations in that form and figure out the coefficients. So, one the one hand I feel that I should have tried it sooner, but on the other I feel that if it were widely known then I would have run across mention of it sooner. Regardless, I am happy to know a little bit more about a set of sequences which I find fascinating (more properties will get their own post one day).
Thursday, August 13, 2009
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.
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.
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.
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