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RE: Re: Bug in multiple integrals with delta function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg90162] RE: [mg90130] Re: Bug in multiple integrals with delta function
  • From: "Tony Harker" <a.harker at ucl.ac.uk>
  • Date: Tue, 1 Jul 2008 06:58:50 -0400 (EDT)

For what it's worth, using Boole gives the correct answer:
g5 = 5!*Integrate[
   Boole[x5 >= x4 && x4 >= x3 && x3 >= x2 && x2 >= x1] 
    DiracDelta[t - x5 + x1], {x1, 0, 1}, {x2, 0, 1}, {x3, 0, 1}, {x4, 
    0, 1}, {x5, 0, 1}, Assumptions -> {0 < t < 1}] 

  Tony Harker 

]-> -----Original Message-----
]-> From: Dr. Wolfgang Hintze [mailto:weh at snafu.de] 
]-> Sent: 30 June 2008 09:53
]-> To: mathgroup at smc.vnet.net
]-> Subject: [mg90130] Re: Bug in multiple integrals with delta function
]-> 
]-> Hello group,
]-> 
]-> I'd like to repeat my question. Could someone please comment.
]-> I'm pretty sure that this is a bug, and what's more:
]-> it's fiendish because it starts to appear only from a 
]-> certain value of a "counter" n.
]-> 
]-> Thanks for your participation.
]-> 
]-> Regards,
]-> Wolfgang
]-> 
]-> "Dr. Wolfgang Hintze" <weh at snafu.de> schrieb im Newsbeitrag 
]-> news:g3d9me$pse$1 at smc.vnet.net...
]-> > Consider the following problem: let x1, x2, ..., xn be 
]-> independent 
]-> > random variables uniformly distributed between 0 and 1, 
]-> and let X = 
]-> > Max(x1, ..., xn) and Y = Min(x1,...,xn).
]-> > The problem consists in calculating the distribution 
]-> function g[n,t] 
]-> > of the variable Z = X-Y.
]-> >
]-> > In trying to solve this problem in Mathematica starting 
]-> with small n 
]-> > and guessing the general formula I found for n>=5 some 
]-> strange and 
]-> > buggy behaviour of a multiple integral containing the function 
]-> > DiracDelta.
]-> >
]-> > I took the following approach
]-> >
]-> > n=2
]-> > g2 = Integrate[DiracDelta[t - Max[x1, x2] + Min[x1, x2]], 
]-> {x1, 0, 1}, 
]-> > {x2, 0, 1}, Assumptions -> {0 < t < 1}] Mathematica 
]-> wouldn't do the 
]-> > integral. Hence I tried to help it by splitting the range of 
]-> > integration thus
]-> >
]-> > Integrate[DiracDelta[t - Max[x1, x2] + Min[x1, x2]], {x1, 
]-> 0, 1}, {x2, 
]-> > x1, 1}, Assumptions -> {0 < t < 1}] 1-t
]-> >
]-> > and
]-> >
]-> > Integrate[DiracDelta[t - Max[x1, x2] + Min[x1, x2]], {x1, 
]-> 0, 1}, {x2, 
]-> > 0, x1}, Assumptions -> {0 < t < 1}]
]-> > 1 - t
]-> >
]-> > which leads to
]-> >
]-> > g2 = 2(1-t)
]-> >
]-> > n=3
]-> > Splitting the range of integration corresponding to 
]-> x1<x2<x3 we note 
]-> > that this would give us 1/3! of the whole range. 
]-> Observing furthermore 
]-> > that now Max = x3 and Min = x1 we have
]-> >
]-> > g3 = 3!*Integrate[DiracDelta[t - x3 + x1], {x1, 0, 1}, 
]-> {x2, x1, 1}, 
]-> > {x3, x2, 1}, Assumptions -> {0 < t < 1}]
]-> > -6*(-1 + t)*t
]-> >
]-> > n=4, similarly
]-> > g4 = 4!*Integrate[DiracDelta[t - x4 + x1], {x1, 0, 1}, 
]-> {x2, x1, 1}, 
]-> > {x3, x2, 1}, {x4, x3, 1}, Assumptions -> {0 < t < 1}]
]-> > -12*(-1 + t)*t^2
]-> >
]-> > We would hence guess the general formula to be
]-> >
]-> > (*)    gk = k(k-1) (1-t) t^(k-2)
]-> >
]-> > But let's continue one step further:
]-> >
]-> > n=5, similarly
]-> > g5 = 5!*Integrate[DiracDelta[t - x5 + x1], {x1, 0, 1}, 
]-> {x2, x1, 1}, 
]-> > {x3, x2, 1}, {x4, x3, 1}, {x5, x4, 1}, Assumptions -> {0 < t < 1}]
]-> > 10*t^2*(6 - 4*t + t^2)
]-> >
]-> > I don't need to bother you further. Since here is the 
]-> bug! According 
]-> > to
]-> > (*) the result should be
]-> > g5ok = 20 (1-t) t^3
]-> > The correct result can also be confirmed by a somewhat different 
]-> > method in Mathematica, viz.
]-> >
]-> > Integrate[Exp[I*w*(Max[x1, x2, x3, x4, x5] - Min[x1, x2, x3, x4, 
]-> > x5])], {x1, 0, 1}, {x2, 0, 1}, {x3, 0, 1}, {x4, 0, 1}, 
]-> {x5, 0, 1}]; 
]-> > Simplify[(1/Sqrt[2*Pi])*InverseFourierTransform[%, w, t], 
]-> 0 < t < 1]
]-> > -20*(-1 + t)*t^3
]-> >
]-> > Hence my question: why does the multiple integral works 
]-> fine for n=2, 
]-> > 3, and 4, but fails for n=5?
]-> >
]-> > Any hint is greatly acknowledged.
]-> >
]-> > Regards,
]-> > Wolfgang
]-> >
]-> > 
]-> 
]-> 
]-> 



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