13 Nov 2000 schoen   » (Master)

Skip to the end of my diary entry, if you'd like.

Arithmetic means and geometric means

Pseudonym, thanks. With your hint, I got an inductive proof.

(All variables are always positive here. We can't necessarily take geometric means of negative numbers, at least if we want to stay in the reals and have a unique solution to everything and ensure that all magnitudes are comparable.)

Here we go: first, to show that, for any two numbers a and b, GM(a,b)<=AM(a,b).

This is equivalent to (ab)^(1/2)<=(a+b)/2, or ab<=((a+b)^2)/4, or 4ab<=a^2+2ab+b^2. Now, that's equivalent to 2ab<=a^2+b^2, which is equivalent to 0<=a^2-2ab+b^2, or 0<=(a-b)^2, which is clearly true. These steps can be reversed to show that GM(a,b)<=AM(a,b). (Note that GM(a,b)=AM(a,b) when, and only when, 0=(a-b)^2, that is, when 0=a-b, or b=a.)

Next, to show that if GM({s=any n numbers})<=AM({s}), then GM({t=any 2n numbers})<=AM({t}).

We observe that GM({s,s'})=GM(GM({s}),GM({s'}) where {s} and {s'} are sets of the same size, and AM({s},{s'})=AM(AM({s}),AM({s'}). This is very easy to see from the definitions of AM and GM. So, if GM({s})<=AM({s}) whenever {s} has exactly n elements, then we can break any set {t} of 2n elements into a pair of sets {s} and {s'} of n elements each (arbitrarily). Then GM({s})<=AM({s}) and also GM({s'})<=AM({s'}). Also, GM(a,b)<=AM(c,d) if a<=b and c<=d (since GM(c,d)<=AM(c,d) and making any element in a set smaller makes the mean, geometric or arithmetic, of that set smaller too). Therefore, GM({s},{s'})<=AM({s},{s'}).

Next, to show that if, for any set {s}, some number x<=AM({s},x), then x<=AM({s}). If x<=AM({s},x), then if {s} contains (n-1) elements, nx<=(n-1)AM({s})+x (since the sum of the elements of {s} equals (n-1)AM({s}). Then nx-x<=(n-1)AM({s}), or (n-1)x<=(n-1)AM({s}), or x<=AM({s}).

Finally, given that GM({t})<=AM({t}) for any set t of n elements, to show that GM({s})<=AM({s}) for any set s of n-1 elements. Let {s} be a given set of n-1 elements. Then take {t} as the set {s} with the addition of the new element GM({s}). Then we have {t} with n elements, so that GM({t})<=AM({t}), by hypothesis. Now GM({s},GM{s})<=AM({s},GM{s}). But clearly GM({s},GM{s})=GM({s}), since adding a new element to a set equal to the mean of the set will not change the mean. Therefore, GM({s})<=AM({s},GM{s}). But we have shown above that this implies that GM({s})<=AM({s}).

By induction, then, the geometric mean of any set is less than or equal to the arithmetic mean of that set.

I still wonder if my binomial thing would work out; there's so much factoring after the binomial expansion. It gets really messy, and it's hard to see whether or not it's going anywhere.

Solar thermal power generation

I spent a lot of time on Sunday wondering about and then reading and writing about solar thermal power plants, an alternative to photovoltaic cells for solar power. If you have lenses or mirrors to focus the sun's rays on a particular point, and you have a good absorber at that location, you should be able to achieve temperatures like the temperature inside the combustion chamber of an engine, but without burning any fuel. I heard a rumor about a power plant design on this principle a few years ago, but I never really looked into it until today.

Given that the sun provides around 1 kW/m^2 of energy at the Earth's surface (depending where and how you measure, and whether you're just talking about visible light or other things too), you'd think that you could get 1 MW/1000 m^2 (which is the area of a square whose sides are about 31 meters). But photovoltaic cells are notoriously inefficient, even though they have been getting a lot better.

It might be that if you used that 1 MW/1000 m^2 to heat some target, which then, for example, boiled water for a steam engine, you might be able to convert the power with relatively high efficiency -- especially since efficient commercial heat engines, including large external combustion engines, have been in production for many generations and are being refined all the time. Semiconductors, and especially light-sensitive semiconductors, are still extremely new. (Imagine what semiconductors would be like if they were as mature as heat engines!)

So, for example, why should we power electrical power plants by burning coal in a combustion chamber, when we could have an array of mirrors concentrating sunlight on the same chamber to raise it to a comparable temperature? Engines in general just rely on steep temperature gradients over long periods of time in order to generate useful power; those temperature gradients can be provided by the absorption of concentrated solar radiation, relative to the cooler outside world.

So, it turns out that there are a lot of people actively working on solar thermal power plants today, including a consortium in Israel which is trying to build a 250 kW prototype plant. The U.S. Department of Energy actually has a test facility at the Sandia National Laboratory in Albuquerque, NM, where you can bring a solar thermal engine for testing -- they rent it out to the public if you have something you want to try! So you put your engine up on a tower or something, and they have a field full of mirrors in the New Mexico desert that all reflect solar energy onto your engine, and you see what happens.

There are a lot of other projects like this, too. I'd like to get some information from some of the people who are working on these things, and maybe go and see one.

I wrote a comparison of solar thermal power plants to other kinds of electrical power plants, showing salient pros and cons of each. The solar thermal plants seem to do really well, but they need a fair amount of research in order to be made cheap enough to be appealing to large energy companies. But the "no fuel, no emissions" bit is a big plus, and solar thermal power beats out hydroelectric power for environmental impact.

I wonder whether a practical solar thermal generator can be built on a small scale, for example as a replacement for a Diesel-powered backup generator. I know that some people are already using solar power to provide all their home energy needs, but most of those are using photovoltaics, or solar thermal heating but not solar thermal electricity generation.

But I think this is really cool stuff.

In other news

I learned a bit more about HTML parsing in connection with some Peacefire discussions. I need to meet some of those folks face to face; I hope we'll all make it to Computers, Freedom, and Privacy 2001 in Cambridge, MA.

Lots of things are still coming up.

It's now a month after the party I went to, um, a month ago.

I was re-reading some Catullus in Latin and English. Still very powerful stuff. Of course, all I'm really familiar with is the Lesbia poems.

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