31 January 2009
Quadratic equations
How many solutions does a typical quadratic equation have? Discuss.
(I'm deliberately leaving the problem vague, because it's more interesting that way.)
(I'm deliberately leaving the problem vague, because it's more interesting that way.)
28 January 2009
Fields medal birthdate quirk?
A comment at this post from the Secret Blogging Seminar, signed "estraven", said that since the Fields Medal is awarded every four years, and only to people under 40, birthdate modulo 4 is relevant.
There's an easy explanation, if true -- assuming that people get something done in their late thirties, a 39-year-old is more likely to have done Fields-worthy work than a 36-year-old. It's our version of the effect that Malcolm Gladwell talked about in Outliers: The Story of Success. There, he points out that the cutoff for most junior hockey leagues in Canada is January 1, and as a result players born earlier in the year are more likely to be selected for teams since they're older than their competition; thus they get better instruction and more practice, and as a result a surprisingly large proportion of players in the highest-level leagues are born early in the year. Something like two-thirds of players in the highest Canadian junior hockey leagues are born between January and April. (You wouldn't see this in the NHL, I suspect, because not all their players are from Canada.)
But I don't feel like doing the research to determine if birthdate modulo 4 actually has an effect on Fields Medal winning. If you feel like it, I'd like to know the results.
There's an easy explanation, if true -- assuming that people get something done in their late thirties, a 39-year-old is more likely to have done Fields-worthy work than a 36-year-old. It's our version of the effect that Malcolm Gladwell talked about in Outliers: The Story of Success. There, he points out that the cutoff for most junior hockey leagues in Canada is January 1, and as a result players born earlier in the year are more likely to be selected for teams since they're older than their competition; thus they get better instruction and more practice, and as a result a surprisingly large proportion of players in the highest-level leagues are born early in the year. Something like two-thirds of players in the highest Canadian junior hockey leagues are born between January and April. (You wouldn't see this in the NHL, I suspect, because not all their players are from Canada.)
But I don't feel like doing the research to determine if birthdate modulo 4 actually has an effect on Fields Medal winning. If you feel like it, I'd like to know the results.
27 January 2009
Universality theory for cranks
From the geomblog, in 2004: a meta-proof of P=/!=NP. With very slight modifications this could be a meta-proof of the Riemann hypothesis, or any other outstanding open problem in mathematics, theoretical CS, theoretical physics, or other heavily mathematical fields. The cranks work in roughly the same way regardless of the specific question.
26 January 2009
Probabilistic fun with the n-sphere
On reddit: a link to the old chestnut that high-dimensional spheres are weird. If you consider the volume of the unit ball in Rn as a function of n, it increases up to n=5 and then decreases. The volume is πn/2/((n/2)!). (By the way, I generally use the factorial notation, not the Γ notation, even when the argument isn't an integer.)
But I hear you complaining that it doesn't make sense to compare volumes in different dimensions! Fair enough. Compare the volume of the unit ball in Rn to the cube circumscribing it, which has volume 2n. Then the portion of the cube which is inside the ball is f(n) = πn/2/((n/2)! 2n). This is rapidly decreasing with n. For n = 2, it's π/4 -- the volume of the unit disc is π, and it can be inscribed in a square of area 4. In n = 3, the unit ball has volume 4π/3 and it's inscribed in a cube of size 8, so we get f(3) = π/6. But f(n) decreases superexponentially. f(10) is about 0.0025, f(20) is about 25 in a billion.
I was surprised that it decayed that quickly -- I'd never bothered to work it out. But if you think about it probabilistically, it kind of makes sense. Namely, a random point in the unit n-cube can be identified with its coordinates (x1, x2, ..., xn). It's in the unit n-sphere if and only if the sum of the squares of those coordinates is less than 1. Let yi = xi2 -- then yi has mean 1/3 and variance 4/45. (That's calculus.) So the sum of the squares of the coordinates is a sum of n such independent random variables, and is thus itself a random variable with mean n/3 and variance 4n/45 -- it's no surprise most of its mass is at n > 1. One could probably use large deviation inequalities to quantify this, but come on, it's 11 at night and I have real work to do.
But I hear you complaining that it doesn't make sense to compare volumes in different dimensions! Fair enough. Compare the volume of the unit ball in Rn to the cube circumscribing it, which has volume 2n. Then the portion of the cube which is inside the ball is f(n) = πn/2/((n/2)! 2n). This is rapidly decreasing with n. For n = 2, it's π/4 -- the volume of the unit disc is π, and it can be inscribed in a square of area 4. In n = 3, the unit ball has volume 4π/3 and it's inscribed in a cube of size 8, so we get f(3) = π/6. But f(n) decreases superexponentially. f(10) is about 0.0025, f(20) is about 25 in a billion.
I was surprised that it decayed that quickly -- I'd never bothered to work it out. But if you think about it probabilistically, it kind of makes sense. Namely, a random point in the unit n-cube can be identified with its coordinates (x1, x2, ..., xn). It's in the unit n-sphere if and only if the sum of the squares of those coordinates is less than 1. Let yi = xi2 -- then yi has mean 1/3 and variance 4/45. (That's calculus.) So the sum of the squares of the coordinates is a sum of n such independent random variables, and is thus itself a random variable with mean n/3 and variance 4n/45 -- it's no surprise most of its mass is at n > 1. One could probably use large deviation inequalities to quantify this, but come on, it's 11 at night and I have real work to do.
Obama's Erdos number
Does anybody know what Barack Obama's Erdos number is? (In particular, is it even defined?)
I learned a few days ago that if the links are people who know each other, my distance from Barack Obama is at most four -- Joe Biden went to high school with the father of a friend of mine. (Yes, this only proves that I'm at distance at most three from Biden; proving that I'm at distance at most four from Obama is left as an exercise for the reader.
Then again, I'm not sure if Obama even has academic publications, or where I'd look to find out if he did. The relevant Wikipedia article says "he published no legal scholarship", citing this New York Times article, though.
I learned a few days ago that if the links are people who know each other, my distance from Barack Obama is at most four -- Joe Biden went to high school with the father of a friend of mine. (Yes, this only proves that I'm at distance at most three from Biden; proving that I'm at distance at most four from Obama is left as an exercise for the reader.
Then again, I'm not sure if Obama even has academic publications, or where I'd look to find out if he did. The relevant Wikipedia article says "he published no legal scholarship", citing this New York Times article, though.
25 January 2009
Existence of bijections?
I have two counting sequences that I know to be the same. That is, I have objects of type A and of type B, which each have a size, and I know that the number of A-objects of size n and the number of B-objects of size n are the same. But I am struggling to find a bijection between the things they count.
Are there results of the form "sure, the two counting sequences are the same, but there's no `natural' bijection between the corresponding sets?" (Obviously bijections will always exist between sets of the same size, so this would of course depend on the definition of the word "natural" -- perhaps a meaning analogous to the category-theoretic one is what I'm looking for, perhaps not.) I've never seen any, but most of the people I know are more interested in finding the bijections than in proving there aren't any.
Are there results of the form "sure, the two counting sequences are the same, but there's no `natural' bijection between the corresponding sets?" (Obviously bijections will always exist between sets of the same size, so this would of course depend on the definition of the word "natural" -- perhaps a meaning analogous to the category-theoretic one is what I'm looking for, perhaps not.) I've never seen any, but most of the people I know are more interested in finding the bijections than in proving there aren't any.
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