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sniffnoyActually I do recall seeing somewhere someone define "finite" to mean "finite nonempty". That seems like a bad idea[0]. Also, this is not actually going to be an empty entry because a.) you are seeing words in it right now and b.) I already did that once.
But I've forgotten most of what I intended to write down in the "finite entry", so, that isn't happening.
Well, one thing I remember. Played a Flash game called Refraction, it's intended for teaching kids fractions, so it starts off ridiculously easy, but it gets fun eventually, and actually getting all the cards is a good challenge. (Also, the whole "for kids" thing seems to have resulted in some annoying interface choices.) (Note, if you play it - if there are multiple coins on a level, it is possible to get them all at once. But if there are coins and a card, it may not be possible to get both the coins and the card at once.)
Anyway, one thing that bothered me was that there's this tile you can use to add fractions - but only if they have the same denominator! Otherwise, you need an expander tile to un-reduce the fraction before adding. (The sum will still keep that same denominator, too, it won't auto-reduce.) Now ordinarily I fall very far to the gamist side of things, and from that perspective it's kind of a neat mechanic that adds additional challenge. But it still bugs me because this doesn't correspond to anything mathematically! They said these beams were fractions, but now I have to make a distinction between a 1/2 beam and a 2/4 beam, while still recognizing that these are numerically equal? Yuck. (Note that the tiles you're trying to get the beams to may have multiple inputs, and in this case you can just stick differently-denominatored beams in each input, and it'll add automatically; and since these tiles don't emit anything, there's no need to worry about what the denominator of the reuslt will be.) Also, the instructions don't specify what happens if an un-reduced fraction goes through a splitter, which actually caused a problem for me at one point (it turns out, it will divide the numerator if possible, and multiply the denominator if not).
Here's something else I remember: Quantification of residual finiteness. The University of Michigan has a "math club" for undergrads which has lectures in the common room every Thursday, and sometime it's something really interesting and a bunch of us grad students show up too. :) So a few weeks ago Khalid Bou-Rabee, a postdoc here, spoke about "quantification of residual finiteness", which is apparently known more formally as "normal Farb growth". Actually he lost most of the undergrads with this one, but it was pretty neat. :)
What it is, is this: Say we have a finitely generated group G which is residually finite, which means every nontrivial element can be detected by some finite quotient. So let's let D(g) mean the size of the smallest quotient in which g is nontrivial, and F(n) be the maximum of D over all nontrivial elements of length at most n. Of course this depends on the generating set, but not if we are just considering the overall growth rate of F.
So he tells us, F(n) is at most logarithmic iff G is virtually abelian. Any (virtually) nilpotent group has F at most polylogarithmic, but the converse is open. So what are some non-polylog examples? Free groups are somewhere between n^(2/3) and n^3, apparently, and the SL_k(Z) are also known to be polynomial. So is it always at most polynomial, then? No! The Grigorchuk group, despite being of intermediate growth, actually has exponential normal Farb growth! I asked him afterward if there was *any* known bound on what sort of function it could be, and he said he thought there might be some tower-of-exponentials-bound but he didn't remember offhand.
...actually, come to think of it, that is most of what I intended to write in the "finite entry". Indeed come to think of it I remember what the third thing was and I think I'll skip it. Guess this is the finite entry after all. The other stuff I intended to write now... I'll do that when I get around to it. (Man, two weeks used to be a *long* time for me to go without posting...)
-Harry
[0]I've mentioned the time that one professor (too tired to recall his name right now) decided he would just use "ideal" to mean "proper ideal", haven't I? Because, he said, otherwise he would forget to say "proper" whenever he meant "proper ideal". To noone's surprise but his own, this just made everything much worse.
But I've forgotten most of what I intended to write down in the "finite entry", so, that isn't happening.
Well, one thing I remember. Played a Flash game called Refraction, it's intended for teaching kids fractions, so it starts off ridiculously easy, but it gets fun eventually, and actually getting all the cards is a good challenge. (Also, the whole "for kids" thing seems to have resulted in some annoying interface choices.) (Note, if you play it - if there are multiple coins on a level, it is possible to get them all at once. But if there are coins and a card, it may not be possible to get both the coins and the card at once.)
Anyway, one thing that bothered me was that there's this tile you can use to add fractions - but only if they have the same denominator! Otherwise, you need an expander tile to un-reduce the fraction before adding. (The sum will still keep that same denominator, too, it won't auto-reduce.) Now ordinarily I fall very far to the gamist side of things, and from that perspective it's kind of a neat mechanic that adds additional challenge. But it still bugs me because this doesn't correspond to anything mathematically! They said these beams were fractions, but now I have to make a distinction between a 1/2 beam and a 2/4 beam, while still recognizing that these are numerically equal? Yuck. (Note that the tiles you're trying to get the beams to may have multiple inputs, and in this case you can just stick differently-denominatored beams in each input, and it'll add automatically; and since these tiles don't emit anything, there's no need to worry about what the denominator of the reuslt will be.) Also, the instructions don't specify what happens if an un-reduced fraction goes through a splitter, which actually caused a problem for me at one point (it turns out, it will divide the numerator if possible, and multiply the denominator if not).
Here's something else I remember: Quantification of residual finiteness. The University of Michigan has a "math club" for undergrads which has lectures in the common room every Thursday, and sometime it's something really interesting and a bunch of us grad students show up too. :) So a few weeks ago Khalid Bou-Rabee, a postdoc here, spoke about "quantification of residual finiteness", which is apparently known more formally as "normal Farb growth". Actually he lost most of the undergrads with this one, but it was pretty neat. :)
What it is, is this: Say we have a finitely generated group G which is residually finite, which means every nontrivial element can be detected by some finite quotient. So let's let D(g) mean the size of the smallest quotient in which g is nontrivial, and F(n) be the maximum of D over all nontrivial elements of length at most n. Of course this depends on the generating set, but not if we are just considering the overall growth rate of F.
So he tells us, F(n) is at most logarithmic iff G is virtually abelian. Any (virtually) nilpotent group has F at most polylogarithmic, but the converse is open. So what are some non-polylog examples? Free groups are somewhere between n^(2/3) and n^3, apparently, and the SL_k(Z) are also known to be polynomial. So is it always at most polynomial, then? No! The Grigorchuk group, despite being of intermediate growth, actually has exponential normal Farb growth! I asked him afterward if there was *any* known bound on what sort of function it could be, and he said he thought there might be some tower-of-exponentials-bound but he didn't remember offhand.
...actually, come to think of it, that is most of what I intended to write in the "finite entry". Indeed come to think of it I remember what the third thing was and I think I'll skip it. Guess this is the finite entry after all. The other stuff I intended to write now... I'll do that when I get around to it. (Man, two weeks used to be a *long* time for me to go without posting...)
-Harry
[0]I've mentioned the time that one professor (too tired to recall his name right now) decided he would just use "ideal" to mean "proper ideal", haven't I? Because, he said, otherwise he would forget to say "proper" whenever he meant "proper ideal". To noone's surprise but his own, this just made everything much worse.
