Posted in GILA, number theory, tagged primes on October 15, 2010 |
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Another small example I noticed awhile ago and forgot to write up.
Prime numbers, as one of the most fundamental concepts in mathematics, have a way of turning up in unexpected places. For example, the life cycles of some cicadas are either or years. It’s thought that this is a response to predation by predators with shorter life cycles; if your life cycle is prime, a predator with any shorter life cycle can’t reliably predate upon you.
A month or so ago I noticed a similar effect happening in the card game BS. In BS, some number of players (usually about four) are dealt the same number of cards from a standard deck without jokers. Beginning with one fixed card, such as the two of clubs, players take turns placing some number of cards face-down in the center. The catch is that the players must claim that they are placing down some number of a specific card; Player 1 must claim that they are placing down twos, Player 2 must claim that they are placing down threes, and so forth until we get to kings and start over. Any time cards are played, another player can accuse the current player of lying. If the accusation is right, the lying player must pick up the pile in the center. If it is wrong, the accusing player must pick up the pile in the center. The goal is to get rid of all of one’s cards.
I’ve been playing this game for years, but I didn’t notice until quite recently that the reason the game terminates in practice is that , the number of types of cards in a standard deck, is prime. If, for example, we stopped playing with aces and only used types of cards, then a game with people need not terminate. Consider a game in which Player 1 has only cards , Player 2 has only cards , Player 3 has only cards , and Player 4 has only cards , and suppose that Player 1 has to play threes at some point in the game. Then no player can get rid of their cards without lying; since the number of players divides the number of card types, every player will always be asked to play a card they don’t have. Once every player is aware of this, every player can call out every other player’s lies, and it will become impossible to end the game reasonably.
More generally, such situations can occur if is replaced by a composite number such that the number of players is at least the smallest prime factor of . This is because people who get rid of their cards will leave the game until the number of players is equal to the smallest prime factor of , at which point the game may stall. But because is prime, any game played with less than people has the property that each player will eventually be asked to play a card that they have.
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Posted in GILA, tagged estimation on September 28, 2010 |
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A small example, but I thought it was funny.
I am currently at Logan waiting for my flight to Heathrow. An hour or so ago, one of my friends asked me how long my flight is. I knew that my flight would depart at about 7:30pm and arrive at about 7:30am, but both times are local. So the actual length of the flight is about 12 hours minus the time difference – which I didn’t know!
But then I realized I could compute the time difference because I knew two other things – the average ground speed of a commercial airplane, and the circumference of the Earth. The average ground speed of a commercial airplane is about miles per hour, which I know from idly staring at that one channel that monitors the airspeed of a plane. The circumference of the Earth is kilometers (to an accuracy of better than one percent!), which I know from preparing for the Fermi Questions event at Science Olympiad. (This is a very handy number to know for certain types of estimates, such as this one.) Given this number, it follows that the velocity of the surface of the Earth is about kilometers per hour, or about miles per hour.
Now, suppose the flight takes hours. Then I have traveled a distance of miles, but at the same time I have crossed approximately time zones. So the difference in local times should be approximately . Setting this equal to and rounding to the closest integer, it follows that the time difference between Boston and London is hours (which it is) and that my flight will take hours (which it will).
An interesting idea this computation illustrates is that if you can estimate an integer (in this case, the number of time zones my flight will cross) with enough precision, you know it exactly. A more sophisticated variant of this idea is that a continuous function from a connected space to a discrete space must be constant.
(Full disclosure: I messed up the last step when I did this calculation the first time.)
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The orbit-stabilizer theorem implies, very immediately, one of the most important counting results in group theory. The proof is easy enough to give in a paragraph now that we’ve set up the requisite machinery. Remember that we counted fixed points by looking at the size of the stabilizer subgroup. Let’s count them another way. Since a fixed point is really a pair such that , and we’ve been counting them indexed by , let’s count them indexed by . We use to denote the set of fixed points of . (Note that this is a function of the group action, not the group, but again we’re abusing notation.) Counting the total number of fixed points “vertically,” then “horizontally,” gives the following.
On the other hand, by the orbit-stabilizer theorem, it’s true for any orbit that , since the cosets of any stabilizer subgroup partition . This immediately gives us the lemma formerly known as Burnside’s, or the Cauchy-Frobenius lemma, which we’ll give a neutral name.
Orbit-counting lemma: The number of orbits in a group action is given by , i.e. the average number of fixed points.
In this post we’ll investigate some consequences of this result.
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Now that we’ve discussed group actions a bit, it’s time to characterize them. In this post I’d like to take a leaf from Tim Gowers’ book and try to make each step taken in the post “obvious.” While the content of the proofs is not too difficult, its motivation is rarely discussed.
First, it’s important to note that there is a way to take direct sums or disjoint unions (category theorists would say coproducts) of group actions: given a group acting on two sets , one defines an action on their disjoint union in the obvious way: pick one action or the other. (Disjoint unions differ from unions in the usual sense because we relabel the elements of the sets so that they cannot intersect.) There’s a great reason to do this, and cycle decomposition showed us a special case: every group action is a direct sum or disjoint union of the action on its orbits.
This is the first step toward a structure theorem. Since the group action cannot “mix” between two orbits, it acts “independently” on orbits, and any question we might want to ask about the group action can be answered by looking at each orbit separately. A group action with a single orbit is called transitive, which means that for every in the underlying set there exists such that . So to classify group actions it suffices to classify transitive group actions.
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Sometimes I worry that I should be more consistent or more lenient about the background I expect of my readers. (Readers, I have to admit that I still don’t really know who you are!) Considering how important I think it is that mathematicians value communicating their ideas to non-specialists (what John Armstrong calls the Generally Interested Lay Audience (GILA)), I should probably be putting my money where my mouth is.
So, inspired by the Unapologetic Mathematician, I have decided on a little project: to build up to a discussion of the Polya enumeration theorem without assuming any prerequisites other than a passing familiarity with group theory. Posts in this project, or any subsequent similar projects, will be labeled “GILA.” The general plan is to talk about group actions in general, the orbit-stabilizer theorem, the lemma formerly known as Burnside’s, and generating functions on the way. Much of this discussion can be found in Section 7 of Stanley’s lectures on algebraic combinatorics.
The PET is a very general way to think about questions of the following nature:
- How many ways are there to paint the faces of a cube if we consider two colorings related by some rotation to be the same?
- How many ways are there to paint the beads on a necklace if we consider two colorings related by a rotation of the necklace to be the same?
- How many ways are there to place some balls into some urns if we consider two placements related by a relabeling of the balls to be the same? (In other words, we want the balls to be “indistinguishable.”)
- How many graphs with a fixed number of vertices and edges are there if we consider two graphs related by a relabeling of the vertices to be the same? (In other words, we want the graphs to be “non-isomorphic.”)
The general situation is that we have a (finite) set of slots and another (for now, finite) set of objects we want to put into those slots, which it is useful to think of as a set of colors that we can “paint” the slots. We also have a (finite) group of symmetries that controls when we consider two colorings to be “the same.” To understand this situation, it is first necessary to understand something about group actions.
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