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- From: kyee@bnlux1.bnl.gov (kenton yee)
- Newsgroups: sci.math
- Subject: a more general constraint integral (was:help me, please
- Message-ID: <1992Sep12.194356.21367@bnlux1.bnl.gov>
- Date: 12 Sep 92 19:43:56 GMT
- References: <1992Sep9.135741.4324@bnlux1.bnl.gov> <1992Sep10.195359.4738@cs.brown.edu>
- Organization: Brookhaven National Laboratory, Upton, NY 11973
- Lines: 24
-
- >kyee@bnlux1.bnl.gov (kenton yee) writes:
- ># int dx_1 dx_2 ... dx_N 1 = ??
- >#where 0 < x_i < 1 subject to the condition
- ># x_1 + x_2 + ... + x_N is less than or equal to 1.
-
- 9.4738@cs.brown.edu> dzk@cs.brown.edu (Danny Keren) writes:
- >S1 = {(x1,x2....xn)/xi>=0, x1+x2+...xn<=1}
- >S2 = {(x1,x2....xn)/0<=x1<=x2...<=xn<=1}
- >What you are asking is the volume of S1. Now, look at the following
- >transformation from S1 to S2:
- >(x1,x2,....xn) ----> (x1,x1+x2,x1+x2+x3,...x1+x2+...+xn)
- >it is trivial that it is 1-1 and onto; also, its determinant is 1,
- >so it preserves volume. Its also trivial that the volume of
- >S2 is 1/n! (symmetry considerations) so your integral is equal
- >to 1/n! .
-
- Thanks. Can you extend this to do integrals of the form:
-
- int dx1 dx2 ... dxn 1 = ???
-
- where x1+...+x4 <=1 AND
- x4+...+x8 <=1 AND
- x8+...+x12 <=1 etc... until xn ? (The double occurence
- of x4 and x8 inside two constraints is not a typo.)
-