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Math Help - is there such a thing ??

  1. #1
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    is there such a thing ??

     \int_T \ t^n \phi(t) \, dt  = \left( \int_T \ t \phi(t) \, dx \right)^n

    n is an integer .

    what could  \phi(t) \ be ?
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  2. #2
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     \phi(t) = 0 ? \
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  3. #3
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    Quote Originally Posted by mmzaj View Post
     \int_T \ t^n \phi(t) \, dt  = \left( \int_T \ t \phi(t) \, dx \right)^n

    n is an integer .

    what could  \phi(t) \ be ?
    Are you talking about for all values of n? Or specific values of n? Obviously if it's the last one, every function fits n = 1....

    -Dan
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  4. #4
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    i'm talking about all positive integers .
    my bad
    <br />
\phi(t) = 0  \<br />
    is rather a trivial solution that i'm not interested in
    i know that there will be a family of solutions , i'm interested in this family .
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  5. #5
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    Quote Originally Posted by mmzaj View Post
     \int_T \ t^n \phi(t) \, dt = \left( \int_T \ t \phi(t) \, dx \right)^n

    n is an integer .

    what could  \phi(t) \ be ?
    Consider:

    \int_0^t x^n \phi(x) \, dx = \left( \int_0^t x\,  \phi(x) \, dx \right)^n and differentiate both sides with respect to t:

    t^n \phi(t) = n \left( \int_0^t x\,  \phi(x) \, dx \right)^{n-1} (t \, \phi(t) \, )


    \Rightarrow t^{n-1} = n \left( \int_0^t x\,  \phi(x) \, dx \right)^{n-1}, ~ \phi(t) \neq 0


    \Rightarrow \frac{t\, \phi(t) }{n^{n-1}} = \int_0^t x\,  \phi(x) \, dx .....
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  6. #6
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    nice !! but i was hoping to get  \phi(t) more explicitly ... such as in terms of special functions , or even a series expansion or so ...
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  7. #7
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    Quote Originally Posted by mr fantastic View Post
    Consider:

    \int_0^t x^n \phi(x) \, dx = \left( \int_0^t x\, \phi(x) \, dx \right)^n and differentiate both sides with respect to t:

    t^n \phi(t) = n \left( \int_0^t x\, \phi(x) \, dx \right)^{n-1} (t \, \phi(t) \, )


    \Rightarrow t^{n-1} = n \left( \int_0^t x\, \phi(x) \, dx \right)^{n-1}, ~ \phi(t) \neq 0


    \Rightarrow \frac{t\, \phi(t) }{n^{n-1}} = \int_0^t x\, \phi(x) \, dx .....
    Well, since this contains two such careless mistakes, I though I'd better set the record straight.

    The result in the last line should of course be \frac{t}{n^{1/(n-1)}} = \int_0^t x\, \phi(x) \, dx.

    Then:

    Let k = \frac{1}{n^{1/(n-1)}} = n^{1/(1-n)} and differentiate both sides wrt t:

    k = t \phi(t) \Rightarrow \phi(t) = \frac{k}{t}.
    Last edited by mr fantastic; April 26th 2008 at 09:25 PM.
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  8. #8
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    thanks , but there is something wrong , 'cause  \phi(t) should be independent of n . the equality is required to be true for any positive integer ... so - obviously - \phi(t) should be independent of n .

    here is what i did :

    assume  \phi(t) is smooth and analytic at  t=t_0 , then it can be expanded in terms of a power series . now setting the correct relations on both RH and LH parts , and integrating over T , we end up with something like this :

     \sum^{\infty}_{r=0}\frac{b_r}{r+n+1}= \left(\sum^{\infty}_{r=0}\frac{b_r}{r+2}\right)^{n  }

    n=0,1,2,3 ......


    now the program is to solve for  b_r in general .. so , is this doable ?
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  9. #9
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    Quote Originally Posted by mmzaj View Post
    thanks , but there is something wrong , 'cause  \phi(t) should be independent of n . the equality is required to be true for any positive integer ... so - obviously - \phi(t) should be independent of n .
    [snip]
    I disagree. A solution containing n is perfectly fine ...... I think the only solution you'll find that's independent of n will be the trivial solution \phi(t) = 0 ....
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  10. #10
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    formally - if i'm not mistaken - the problem transforms to finding a set of measure spaces whose measure (  \ ds =\phi(t)dt ) and nth norm <br />
\left\|t\right\|_n = \left( \int \ t^n\ ds \right)^\frac{1}{n}<br />
satisfy :

    1-  \int ds =1

    2- \left\|t\right\|_n= \left\|t\right\|_1 ,  n=2,3,4 ....

    i think the problem went harder , but more formal .
    Last edited by mmzaj; April 28th 2008 at 11:36 AM.
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  11. #11
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    Quote Originally Posted by mr fantastic View Post
    I disagree. A solution containing n is perfectly fine ...... I think the only solution you'll find that's independent of n will be the trivial solution \phi(t) = 0 ....
    i agree , but - and it's a big fat but - the problem in hand requires the equality to hold for every +ive integer . if 0 is the only solution , i need a proof .
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  12. #12
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    here is another equivalent formulation .

    \int \ e^t \phi(t) \, dt = \exp\int \ t \phi(t) \, dt
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  13. #13
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    so , i have been discussing the problem with the guys in physics forums , here is the link
    a problem in Lp spaces .

    i think it helps to look at it .
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