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I am told that I need to solve , and in the textbook example they get a polynomial with only m4, m2 and m0 terms in it, which lets them solve a quadratic in m2, take a square root and compare the results to the range which you are given at the start to find the right one, but I keep getting equations like , where the p.d.f f(x) of the continuous random variable X was the above for , which eventually gets me to , but I don't know how to solve this. In another example, I had the p.d.f. f(y) of the continuous random variable Y being for , which got me to , but, again, I don't know what to do with this. It Is Me Here t / c 08:58, 21 June 2009 (UTC)
Not a massive fan of induction and so I would like to check my solution to a problem with you guys.
"Let . Prove by induction on n that is a polynomial."
Assume that is a polynomial for n=k.
So
Differentiate the above wrt x.
Factorise
is still a polynomial for any f(x), so let us call it g(x).
So we have
So if the result is true for k, it is also true for k+1. Now let n=0 (the question doesn't actually state which set of numbers 'n' belongs to but I assume it's the non-negative integers), which gives us the case where . So by induction, is a polynomial for all integers n, n≥0.
Is that airtight? Thanks. asyndeton talk 14:15, 21 June 2009 (UTC)
Do the rules of double-deck cancellation hearts generalize well to hearts games with N decks and 4N-2 to 4N+2 players? Neon Merlin 23:46, 21 June 2009 (UTC)
Well my first thought is that there are two possible ways in which the game could be generalised, either we stick with the rule that if a second identical card is played then the two cards cancel, which leads to a game where if odd numbers of an identical card are played the last player to play that card (assuming it is the strongest one, otherwise it makes no difference) takes the trick whereas if even numbers are played the cards are ignored, this seems slightly artificial to me and as if it would lead to a confusing game.
Alternatively the other way in which the game could be generalized is to only apply the cancellation rule when all N of an identical card are played in one trick, otherwise if say K (<N) identical cards were played in one trick then the last player to play the identical card would count as having played it and the other players who played it would be ignored. This to some extent removes the excitement that double-deck cancelation hearts has, since especially when N is large the chance of a cancellation occuring are very slim.
Both possibilities have their pros and cons, when N is small (say 3 or 4) I think both variations would make interesting playing. —Preceding unsigned comment added by 86.129.82.211 ( talk) 00:52, 22 June 2009 (UTC)
Mathematics desk | ||
---|---|---|
< June 20 | << May | June | Jul >> | June 22 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
I am told that I need to solve , and in the textbook example they get a polynomial with only m4, m2 and m0 terms in it, which lets them solve a quadratic in m2, take a square root and compare the results to the range which you are given at the start to find the right one, but I keep getting equations like , where the p.d.f f(x) of the continuous random variable X was the above for , which eventually gets me to , but I don't know how to solve this. In another example, I had the p.d.f. f(y) of the continuous random variable Y being for , which got me to , but, again, I don't know what to do with this. It Is Me Here t / c 08:58, 21 June 2009 (UTC)
Not a massive fan of induction and so I would like to check my solution to a problem with you guys.
"Let . Prove by induction on n that is a polynomial."
Assume that is a polynomial for n=k.
So
Differentiate the above wrt x.
Factorise
is still a polynomial for any f(x), so let us call it g(x).
So we have
So if the result is true for k, it is also true for k+1. Now let n=0 (the question doesn't actually state which set of numbers 'n' belongs to but I assume it's the non-negative integers), which gives us the case where . So by induction, is a polynomial for all integers n, n≥0.
Is that airtight? Thanks. asyndeton talk 14:15, 21 June 2009 (UTC)
Do the rules of double-deck cancellation hearts generalize well to hearts games with N decks and 4N-2 to 4N+2 players? Neon Merlin 23:46, 21 June 2009 (UTC)
Well my first thought is that there are two possible ways in which the game could be generalised, either we stick with the rule that if a second identical card is played then the two cards cancel, which leads to a game where if odd numbers of an identical card are played the last player to play that card (assuming it is the strongest one, otherwise it makes no difference) takes the trick whereas if even numbers are played the cards are ignored, this seems slightly artificial to me and as if it would lead to a confusing game.
Alternatively the other way in which the game could be generalized is to only apply the cancellation rule when all N of an identical card are played in one trick, otherwise if say K (<N) identical cards were played in one trick then the last player to play the identical card would count as having played it and the other players who played it would be ignored. This to some extent removes the excitement that double-deck cancelation hearts has, since especially when N is large the chance of a cancellation occuring are very slim.
Both possibilities have their pros and cons, when N is small (say 3 or 4) I think both variations would make interesting playing. —Preceding unsigned comment added by 86.129.82.211 ( talk) 00:52, 22 June 2009 (UTC)