question and answer thread.

I hope this series isn't getting boring yet.:) As I have said I am heading for a proof of the Banach-Tarski paradox and one more piece is needed (actually two, but I am going to assume the existence of two independent rotations of the sphere).

DEFINITIONS

The action of a group G on a set X is called locally commutative if any two elements of G, which have a common fixed point, commute as elements of G.

END OF DEFINITIONS

Proposition: Let C be a subset of a group G acting on X and let I be an index set for a partition of C. If <C> (the group generated by C) is a free subgroup of G acting locally commutatively on X, then each connected component of B(X,XI,CI) contains at most one cycle.

I would proceed by contradiction.

I think its clear to me I'm gonna have to think on this one for a little while. I have lots of little pieces I've pulled out of my toolkit but can't seem to fit the pieces coherently together to convince myself a particular line of reasoning might work. My troubles seem to center on the correct meaning of the group action as a free group or possibly what is exactly the points fixed by an action here (I keep seeing it, then I think of something not quite settling about my reasoning), but if I concentrate real hard on my notes, maybe I'll clear the jungle...

Basically here are some of the things that stand out to me:

It seems the contradiction needs something like:
For a bipartite graph with more than one loop and the given action, find a point whose stabilizer is not commutative.

or possibly:
For ..., show that the a particular element must have finite order (so that <C> is not a free subgroup?)...

The only telling analogy that keeps whispering in the back of my mind is that the fundamental group of a circuit is the free group on one generator (and thus abelian since powers of a generator commute), whereas the fundamental group of the figure-8 is the free group on two generators (not abelian). Since any graph with two circuits either shares a path or has a path that connects two of its vertices, it seems a contraction is possible here. But I don't quite see yet the relation of the fundamental group to any of this yet, except the analogy...

I have this feeling that I am missing something more obvious, but the free group thing is throwing me. Although my definition of the group was intimately tied to the action imposed by the fact that the graph was bipartite and had a perfect matching from which I could pull out the functions, I'm having diificulty entering any of it into a proof...

Caffeine is not helping, no matter how hard it tries...

Well the proof of the result is in the above post...

Remember the key about the group being free is that there are no relations...

For completeness let me provide some discussion on the correspondance between group elements and paths, at least as I see it..

DISCUSSION

Given a graph G(V,C), let <C> denote the free group of all reduced words on the elements of C, so that for all c in C, c and its inverse do not appear adjacent in any element of <C>. In general the elements of <C> have only a formal meaning. However, when each element of C is invertible, i.e. is 1-1 onto its range, then an element of <C> determines a partial function from a subset of V to a subset of V. For instance, when G = Gc(V) is a Cayley graph then <C> is just the group generated by C, where many of the elements of <C> could be the identity element. There is a natural correspondance between paths in G(V,C) and elements of <C>. Suppose P(e1,....,en) is a path in G. For i=1,...,n-1, if ei=<x,y,c> then there are two cases:

1. if y is a vertex of ei+1 then let gi = c,
2. if x is an element of ei+1 and x != y, then let gi = c inverse.

for en = <x,y,c>
1. if x is a vertex of en-1 then let gn=c
2. if y is a vertex of en-1 and x != y, then let gn = c inverse


According to this rule one can associate to the path P(e1,...,en) a unique element gp=g1...gn of the free group <C>. In general this "correspondance" is neither 1-1 nor onto. Note that when G = Gc(V) is a Cayley graph, each group element corresponding to a path in G is reduced in the generators C, and that distinct paths starting at the same point yield formally distinct group elements on the generators C.

Note: If <C> is a free group then they are distinct element.

END OF DISCUSSION

Soooo... maybe I was trying to conceptualize too much the functional foundations of the graph and was unable to let the group just be like the normal homotopy groups I am used to playing around with? Edges, paths, and words, oh my!

Then by

Quote:

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Originally posted by souldog
Well the proof of the result is in the above post...

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you mean my post and particularly the part about the fundamental groups, right?

Something like: pick a point between a choice of two cycles (either on the common path they share or on the path that separates them, as the case may be). Then form the path that goes around one cycle then the next (both, say, clockwise). Form the other path that goes around the latter then the first (still clockwise). The two paths are equivalent in their action, but do not commute on the free group (xy != yx) which violates the hypothesis of local commutivity. Thus the graph cannot have two or more cycles because of the contradiction.

It was really that simple, but I just got lost?

That is exactly it.:)

Now let me tie it all toghether..


The Banach-Tarski Paradox : S2 is (2,2)-paradoxical under the action of SO(3).

proof:

I will assume the well known result that there are two independent rotations of the sphere, which generate a free group of rank two. Since every subgroup of a free group is free, there are free groups of any finite rank. In particular there exist four independent rotations of the sphere {g1,g2,h1,h2}.

It is easy to see that the action of any group G of rotations on the sphere is locally commutative. Each rotation of the sphere fixes exactly the two points of ontersection of its axis of rotaion with the sphere. So if two elements of G share a common fixed point, then they must be about the same axis and so must commute as group elements.

Let B(S2, S2I,CI) be as in the example above, where C={g1,g2,h1,h2}. Now by the second proposition it is enough to see that this graph has a perfect matching. Since <C> is a free subgroup of the rotation group and acts locally commutatively on S2, the last proposition shows that each connected component has at most one cycle. Also each vertex in S2 has valency 4 while each vertex in S2I has valency 2. So by the first proposition there is a perfect matching.

QED.

Lets push it a bit further.

An infinite number of spheres from one:
Having assumed the existence of two independent rotations of the sphere g,h the set {g^nh^n: n=1,2,3,4,..} yields countably many independent rotations. So let C={g1,g2,g3,...} be independent rotations of the sphere. Let I={1,3,5,7...} be the odd integers and for each n in I Cn={gn,gn+1}. Then B(S2, S2I,CI) has one copy of the sphere on one side and countably many on the other.The group <C> is a free subgroup of SO(3) and acts locally commutatively, also the valency of each vertex in B is either 2 or infinite. So the graph contains a perfect matching.
For each gn in C let Agn denote the set of vertices in S2 contsained in an element of the perfect matching colored gn' (the extended function). Then {Agn, Agn+1: n=1,3,5,..} is a partion of S2 and for each n = 1,3,5,... {gn(Agn), gn+1(Agn+1)} is a partition of S2.


Note there is nothing special about the sphere here. The above two results apply to any set acted on by a group G, which contains a free subgroup acting locally commutatively. Also since it has been shown that there exist countably many independent rotations of the sphere one can obtain a continumm of spheres from one using the same proof,

Heck lets keep going

The solid Ball in three space is paradoxical:
proof:

Let me denote the unit ball in three space by B3.

A heck I am tired of typing. See if you can prove this.

Quote:

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Originally posted by souldog
The solid Ball in three space is paradoxical:
proof:

Let me denote the unit ball in three space by B3.

A heck I am tired of typing. See if you can prove this.

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Well, since I learned the traditional decomposition method to prove Banach-Tarski, I can just use that information here :).

You just look at the piece of the ball with 0 < radius < 1. These are paradoxical for the exact same argument as the 2-sphere (at each fixed radius). There is also the polar set of mappings on the boundary 2-sphere which map all of the 2-sphere uniquely except for one point to the plane which has the same free group symmetry structure as the 2-sphere except for that point (which can be isolated). Therefore, this boundary is paradoxical in away that we can use to obtain two points (the original center and the non-uniquely mapped "pole" point) to use as centers for the reconstruction of two balls. These two points are just an added component to the bigraph that this construction shows is paradoxical (and in fact the four components and the boundary show the five component decomposition normally used). I could of course give the actual use of the generators in this decomposition and the resulting recomposition partition, but that is not necessary to just show paradox conditions on the ball (and would just be a copy of what is already out there in the books I learned this from originally)...

I've looked over the discussion so far and rewritten everything using my own notations, and I feel much more secure in my understanding of what is going on now. My early confusion was in not recognizing the role of the bipartite graph (and thus making finitary assumptions or mistaking the functional role). I want to thank you, souldog, as these little exercises have made much clearer in my mind the origins of this fascinating little theorem and in the process shown me a nice little bipartite graph construction which seems to be applicable in many situation where one needs to look at mappings and their domain and codomain in general. I am certainly appreciative!

To use the above methods you just tack a couple of edges which use translations into a connected component without cycles of the bipartite graph formed from the unit ball without the origin. These two new edges put a cycle in this connected component and just correspond to translating some points to the origin. You then just proceed as normal, making sure one of these new edges is in the matching.

What I think is nice about these results is they show how "trivial" the Banach-Tarski paradox really is. What one could do now is add some topology. The functions which determine the graph could be partial hemeomorphisms etc..

DEFINITIONS
Let G be a group of homeomrphisms acting on a topological space X. The set X is said to be generically (m,n)-paradoxical under G, if for some m and n there are disjoint subsets of X A1,...,Am,B1,...,Bn and group elements g1,...,gm,h1,...,hn such that each of {A1,...,Am,B1,...,Bn}, {g1(A1),...,gm(Am)} and {h1(B1),...,hn(Bn)} each partition a comeager subset of X.
END OF DEFINITIONS

You might be interested to know about the following two results.
What I am going to call the

Dougherty-Foreman Paradox: S2 is generically (3,3)-paradoxical under the action of SO(3) using open sets.
The above result is entirely constructive.(No Choice)

And

Theorem: S2 is (3,3)-paradoxical under the action of SO(3) using sets with the property of Baire.


Ok, My Reign Of Terror is over.

Hmmmmmm....

Quote:

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Originally posted by souldog
Dougherty-Foreman Paradox: S2 is generically (3,3)-paradoxical under the action of SO(3) using open sets.
The above result is entirely constructive.(No Choice)

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I had heard about an entirely predicative version, but never knew what to look for. Now I have something new to study (and it goes to show another example of why much of modern math doesn't require choice!).

Ok. Let me change the format a little to something not of this ilk for a little while.

Setting
Its 1983. Seven year old galathaea is sitting in class and listening to the teacher read a story during story-time (I think it was probably the BFG by Dahl). Picture corduroy. Reagan is president. Kids are picking their noses carelessly...

Internal Narrative
A thought is running through galathaea's developing mind, with the autistic self-involvement of youth. Something heard on Mr. Wizard the day before (where, amongst the many wonders of the world, galathaea first learned it was possible to cut a hole in the middle of a 8 1/2 x 11 piece of paper large enough to walk through!). A question is forming...

Action
Suddenly, galathaea raises a hand. The teacher stops reading and looks to galathaea. "Yes?" the teacher asks, expectant.

"If hot air rises and cold air falls, why is it so hot in the deserts and so cold in the mountains?" ignorant little galathaea asks.

Response
The teachers tells galathaea that she will have to go home and ask some friends. When the teacher returns the next day, she tells young galathaea a story so convoluted and full of hand waving, that even young galathaea recognizes it as hand waving and amounted to saying that the air is already cold in high places and gets cooled as it rises.

It took galathaea 6 more years before galathaea finally understood the real reasons, which would have been understandable at the younger age if the teacher had known the real answer.

Repercussions
This was the first time galathaea really noticed that teachers often tell their students false things in school. Throughout the rest of galathaea's scholastic career, this became a hauntingly reoccurring theme that has shaped galathaea to this day. The list of errors conveyed to galathaea through school grew larger and larger each year. Examples of how this problem enter into education are too numerous, but another two examples are indicative.

Recently galathaea was looking over the younger brother's homework (8th grade), and noticed an explanation of tidal forces that mentioned nothing about moments and the fact that the moon was in free fall around the earth (which actually showed why the hand-waving in the book was wrong). Galathaea took the opportunity to explain the idea of forces with moments, and how the forced solid-body movement of the moon around the earth gave rise to pseudo-forces in the well known cross conformation that caused the tide. In the process, galathaea was able to tell bro how this also was one of the two leading reasons for the molten core of the earth(the other being radioactive heating), and how it also gave the moon and all sattelites a force to spin them in the way seen around the solar system (except for the exceptions).

Another example that explores the cause more closely came from a philosophy of science course taken in college. The course had just been entered as an option for education majors to fill one of their requirements, and was thus full of them. The teacher was explaining how certain problems often arise in science that challenge scientist to adjust ore refine their worldview, and an example began to be explained. The choice was Olber's paradox and the challenge it presented to the idea of an infinite, homogeneous, and isotropically populated universe. Suddenly, an education major raised their hand and asked why we were studying actual science (since the teacher went over the math). After a brief recapitulation of the motivation, another ed major chimed in asking if it would be on the test. Suddenly, there were all sorts of objections on the idea that this little excursion didn't fit their list of prerequisites, all (and I mean all) of which came from the education majors. And even though the math was never to be tested in the first place (it was illustrative), the teacher just dropped the example and moved on because too much time had been wasted.

Endgame
This is why galathaea feels so strongly that the future of education lies in interactive computing, most likely on-line, potentially from home. Individual teachers are way too limited in their knowledge, and the lecture/board idiom of teaching is so limited compared to the multimedia and interactive experience available with computers. Things like graph theory and combinatorics should replace things like teaching how to add or subtract at a young age (they are algorithms in number theory after all which are not needed until after the relevant introduction, since we have had things like calculators for a century or so). Kids can manipulate the little sets of objects and learn the basic properties quite easily and intuitively, since many kids already play such games in their heads (I was a maze fanatic, for example). Imagine picking from the best lessons available around the world for a child.

Maybe such an education could finally put an end to incompetent teachers giving their c++ students the void main syndrome...

Reprise
So now I ask you:

If hot air rises and cold air falls, why is it so hot in the deserts and so cold in the mountains?

There is one effect which gives the overwhelmingly largest contribution to this, so it is what I am looking for. Other effects can be mentioned, but focus on the big one.

just a note: the result will be found as Marczewski's problem which was solved by Dougherty and Foreman

Come on guys!!! This one is not a trick question, and I don't expect answers with equations or anything of that sort. I just want an explanation that you could tell a kid that asks this question:

If hot air rises and cold air falls, why is it so hot in the deserts and so cold in the mountains?

I'll give a little hint: This is an on-average stable process. But the "hot air rises" part shows that one part of the process tends equilibrium different than what we see. Driving this stability away from the implied equilibrium must require some other pieces to the explanation. Why is the temperature in mountains cooler than that below? Ther must be some driving source and an explanation for the inversion, so there are at least two small little pieces for this process that give an on-average result different than the idea the hot air above, cool air below you find in, say, a box that is in dynamic equilibrium but hasn't reached thermal equilibrium. Sure its a thermodynamics question, but it can be explained quite simplistically. Where is the heat coming from? Where is it going?

Well in the deserts it's so hot for 2 reasons
1. they are on the equator and the sun ray fall vertivally on them
2. they are low so the air is thick (or dense I dont know the exact word) enough to keep the warmth of the sun whereas on the mauntaing the air is sparse (or someting) and cannot keep the sun's warmth.

Quote:

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Originally posted by Seventh Star
Well in the deserts it's so hot for 2 reasons
1. they are on the equator and the sun ray fall vertivally on them
2. they are low so the air is thick (or dense I dont know the exact word) enough to keep the warmth of the sun whereas on the mauntaing the air is sparse (or someting) and cannot keep the sun's warmth.

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Let's look at these points one at a time.

1. Well, not all deserts are on the equator or even closer to it on average than certain other types of climates. But you have pointed out two important factors: sunlight and latitude. Temperature normally varies on average by latitude due to the amount of sunlight it receives per area. But deserts often have higher temperatures than other climates at the same latitude and elevation, particularly in the summer for that latitude. Where does this higher temperature come from? By explaining this, the explanation for why one commonly finds that deserts in the winter at non-tropical latitudes are often cooler than other climates at that latitude and elevation -- this is not a "counter-example" to the question however, since no such comparison is implied; it is merely another fact to tell a child that asks this...

2. Here, there is the start of something, but it still does not have explanatory power. You are correct that the air at low elevations is on average denser than that at high elevations, but this does not prevent the hot air from rising. In fact, there are regions of the atmosphere that do have this temperature profile where hot air is above cool air, even though density does on average decrease even in those regions. And there are regions in the atmosphere that again go from hotter temperature to lower as one goes higher. So these must have something in common with our layer of the atmosphere; ie. they have a source of heat and a reason for the temperature to decrease as elevation increases. A hint here is that the decrease in temperature does indeed have to do with density, but not for the reason given.

So, it is a start at working towards the solution. But it is not a correct line of reasoning as is...

Here's a shot in the dark: countercurrent diffusion?

Hot air going up passes cold air going down in the opposite direction, allowing for maximum diffusion... but I have a feeling the correct answer has nothing to do with this. Probably because there's less pressure higher up, the warm air can expand, causing it to cool.

1. well the denser air can keep the warmth stored during the day. The ground also. (the sand doesnt do this really good and hense the really cold temperatures in deserts at night). Really high places do not have much ground to keep the warmth (only rocks) and the less dense air also
The ground of the deserts reflects the suns rays and the air manages to keep it. In higher grounds it does not.

2. Oceans and seas (even big rivers) keep the temperature stable in areas near to them (only horisontally but not vertically)

But anyway i think that the ground temperatyre plays the most important role followef by the density of air.

Quote:

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Originally posted by solarflare
but I have a feeling the correct answer has nothing to do with this. Probably because there's less pressure higher up, the warm air can expand, causing it to cool.

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Quote:

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Originally posted by SeventhStar
The ground also. (the sand doesnt do this really good and hense the really cold temperatures in deserts at night). Really high places do not have much ground to keep the warmth (only rocks) and the less dense air also
The ground of the deserts reflects the suns rays and the air manages to keep it. In higher grounds it does not.

2. Oceans and seas (even big rivers) keep the temperature stable in areas near to them (only horisontally but not vertically)

But anyway i think that the ground temperatyre plays the most important role followef by the density of air.

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And here we have the outline for the basic answer to the question. Now I'll try to toss it together in an explanation that could be given to a kid who asks this that would be useful to get a start at understanding what is going on:

Where does the heat come from?
Its warmer during the day than it is at night. So its pretty clear that the heat around us is because of the sun. It might be thought (by this hypothetical kid) that because the sun is really hot, it makes the Earth warm just like when you stand close to a hot stove you get warm. The sun is really hot, but for something really hot to heat another thing the way a stove does requires there to be something in between the two things to carry the heat. A stove does this through the air, but space doesn't have as much stuff in it, and not much heat can move from the sun to the Earth this way. However, the sun is also very bright, and the sun makes the Earth warm by shining light on it. Light and heat are two different forms of the same thing: energy. When light shines on things, some of it gets bounced off of it (like a mirror), some of it goes right through it, and some of it gets absorbed. A window, for example, lets you see alot of the light that shines through it. It is the absorption that is the key here.

When light gets absorbed by something, what is actually happening is that the light energy changes into a type of energy that makes the small pieces of the thing (the atoms, if the kid knows about them) start to move around alot, since moving is a kind of energy too. These atoms wiggle around faster as they absorb the light, and this wiggling is what we call heat. The hotter an object is, the more these things wiggle.

So, alot of the light from the sun just passes through the air and hits the ground (some of it gets absorbed and some reflected -- and this can be explained to the child as well). You can't see through ground very well (:)) so it must be absorbing or reflecting much of the light. The light that is absorbed heats the ground, and that is why the ground gets warm and where a lot of the heat on the ground comes from (of course, there is internal heating from the tidal forces and radioactive decay as I think I mentioned in my question backgrounder, but it is not the main source of heat -- this too can be told to the kid).

Why is the desert ground hotter than the ground on a forest?
Things are held together tightly by these things called forces. Even though all the pieces (atoms) of stone are wiggling around really fast, the stone doesn't fly apart because these forces hold it together. But different types of things have different amounts of force holding it together. (An explanation to the kid could point out that atoms and molecules are different for different things, and each different type of atom or molecule pulls the other atoms and molecules to it in different strengths). Since these pieces are held in place with different strengths, some things need more energy than others to cause them to wiggle around as much as other things (a small introduction to momentum and inertia could fit in here as well, but it is not as big a factor as the binding forces). This is a big point, because the thing we call temperature is not the amount of energy they have absorbed (which we call heat) but how fast they wiggle around. So some things get hot much faster than other things (a great place to start an introduction to specific heat).

Water gets heated slower than most other things around us. Although it is a liguid and moves around a lot, and it sure seems easier to pull it apart than a rock, water has some really strong forces in it keeping it together (hydrogen bonding could eventually be explained, but that should wait), and rocks normally let their atoms or molecules wiggle around alot faster without letting them break out of their place. So places that have a lot of water tend to warm up slower as they absorb the sun's light. Deserts are places that don't have a lot of water (eventually again the kid can be taught why deserts don't have a lot of water, telling them about mountains, rain shielding, and all). So the ground gets warm very quickly in the desert. In forests, and any place where there is a lot of plants, there is a lot of water (since plants need the water, they grow best where there is a lot of it).

The temperature of the ground warms the air that touches it, and if they are touching for a long time, they tend to reach the same temperature (this is the equilibrium law of thermodynamics). So the air around the ground in deserts tends to be around the same temperature as the hot ground, and the air around the ground in wetter places tends to be the same somewhat cooler temperature as that place's ground. It is the temperature hear that gets shared, not the amount of energy or heat there.