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        riddles >> putnam exam (pure math) >> Blending Water
        
(Message started by: Michael_Dagg on Dec 16^th, 2006, 10:41am)

Title: Blending Water
Post by Michael_Dagg on Dec 16^th, 2006, 10:41am

Let W be a blender of water containing n molecules of H_2 O.
Turn on the blender and let it run as long as you like and then turn
it off. Discover a way to show that there is at least one molecule m_k
that is in the exactly same position it was before the water was blended.

Title: Re: Blending Water
Post by Icarus on Dec 16^th, 2006, 5:38pm

Umm... The direct physical interpretation is obviously false, and there are a number of possible mathematical models depending on how you define the word "position". I am particularly confused by the apparent blending of continuous and discrete.

Could you be a little more explicit about the mathematical model you intend?

Title: Re: Blending Water
Post by Whiskey Tango Foxtrot on Dec 16^th, 2006, 8:04pm

Also, what does m_k mean here?

Title: Re: Blending Water
Post by physx on Dec 16^th, 2006, 11:46pm

This seems a direct application of the Banach Fixed Point theorem. You can state this problem in another way by supposing you have two identical pieces of paper superimposed on each other. Then, you take the top piece and crush it into a ball. You can prove there is a point on the crushed ball that is exactly on top of the point it was before it was crushed. This happens because crushing (or blending) is the same as mapping a function into another function. I've also seen this problem stated in yet another way: You observe a coconut, covered in its fibers. You can prove that there will always be at least one whorl of the fibers on the surface of the coconut. I'd really like to see other concrete applications to the fixed point theorem (mind you, there are many: http://mathworld.wolfram.com/FixedPointTheorem.)

Title: Re: Blending Water
Post by Michael_Dagg on Dec 17^th, 2006, 2:33pm

m_k is just some kth modecule.

I have a lot more to say about this problem I don't have much
time at this moment to say more because I have to go somewhere
but I will say that it is true for n >= 1, and discretely that is (but this
is a trivial case) -- it is discoverable, however.

The two-sheets of paper analogy is certainly related but the top sheet
is a finitely multiply punctured domain as well and so it too in general is a
discrete domain (I am not saying that a punctured domain is necessarily
dscrete of course). There a couple of ways to get around this and really
they are simple.

A Banach space is not required here, i.e. so-to-speak, as there are
domains that are Bananch space naturally (i.e. any Hilbert space works
for this same problem, triivially nevertheless).

Title: Re: Blending Water
Post by Icarus on Dec 17^th, 2006, 7:54pm

The problem with the fixed point theorem in this application is that he is discussing discrete molecules which, on that scale, are each surrounded by empty space. It is true that you can model the blending with a continuous function of which the molecules represent discrete points. But the fixed point thus guaranteed has 0 probability of happening to be at a point representing a molecule. If we represent the molecules as having larger than 0 volume, the probability is greater than 0, but is still not 1 unless we are unrealistic in our model in the opposite direction. And also in this case, can we say that a molecule is in the exact same position if it's orientation has changed?

As for why I said that the direct physical interpretation is clearly false, all I need is one axis though the water on which no molecule falls (which - since matter is mostly empty space - is almost guaranteed to exist), and I can rotate about that axis, moving every molecule to a new position.

Therefore, I must assume that either this problem is not meant to be interpreted discretely despite its reference to indexed molecules, and we really are meant to apply a fixed point theorem to obtain a point that does not move. Or else, we are intended to discretize space in some fashion, and the blending represents a rearrangement of the molecules in their various discrete spacial cubbyholes (which I doubt, as you can just move every molecule into the previous cubbyhole of the next higher molecule - with the highest numbered molecule moving into the 1st).

Hence my confusion, and request for clarification.