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Please post a problem

  1. Aug 13, 2006 #1


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    Would someone please post a good problem (or at least an interesting one) for me to work on, say calculus, real/complex analysis, or some generating function stuff, or some problem anybody can understand but will scratch their head at? I really bored. Thanks for cherring me up,

  2. jcsd
  3. Aug 13, 2006 #2


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    If P is a polygon, prove that it cannot be the union of disjoint convex quadrilaterals, each of which has exactly one face that is also a face of P.
    Last edited: Aug 13, 2006
  4. Aug 14, 2006 #3
    A particle is moving on the line y=x^3 in the first quadrant starting from Origin at a speed dy/dx=1 unit per...time unit. ignore units. :p

    Question is...find a formula for the angle formed by the x-axis and a line created by the Origin and the particle's position.

    In all honesty I saw it before but I never took the time to solve it. I doubt i can however ... :(
  5. Aug 14, 2006 #4


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    Find the sum:

    \sum_{i = 0}^{+\infty}
    \tan^{-1} \frac{1}{1 + x + x^2}

    (I think I have that right)
  6. Aug 14, 2006 #5
    Quite easy: it is a diverging series ...
  7. Aug 14, 2006 #6


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    Hurkyl, do you mean [tex]\sum_{x = 0}^{+\infty}\tan^{-1} \frac{1}{1 + x + x^2}[/tex] ?
  8. Aug 14, 2006 #7


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    Yes, that one looks better!
  9. Aug 16, 2006 #8


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    [tex]\sum_{x = 0}^{+\infty}\tan^{-1} \frac{1}{1 + x + x^2} = \sum_{x = 0}^{+\infty}\tan^{-1} \frac{(x+1)-x}{1 + (x+1)x} = \sum_{x = 0}^{+\infty}\left( \tan^{-1}(x+1)-\tan^{-1}x\right) = \lim_{M\rightarrow\infty} \tan^{-1}(M+1)-\tan^{-1}(0)= \frac{\pi}{2}[/tex]

  10. Aug 16, 2006 #9
    Try with this.

    Find the product

    [tex]\prod_{k=1}^{\infty} \left( 1 + \frac{2}{k^2 + 7} \right)[/tex].
    Last edited: Aug 16, 2006
  11. Aug 17, 2006 #10
    Well, I have a problem which currently bugs me (although I think I already solved it). Find the solution to the differential equation:

    dv/dt = a*v+b*v^2.

    It represents the movement of a particle with a velocity dependent friction force (which is proportional to v for small v and proportional to v^2 for large v). So the solution must be equal to the stokes case for small values of v and equal to the newton case for large values of v, that´s how you can check if your answer is correct.

    I made the problem up myself, so it might not be physically correct. But at least it´s a solvable DE which you can make into a linear first order ODE by substitution. Sorry for my english.
    Last edited: Aug 17, 2006
  12. Aug 24, 2006 #11


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    Since [tex]\frac{\sin \pi x}{\pi x}=\prod_{k=1}^{\infty} \left( 1-\frac{x^2}{k^2}\right) = \prod_{k=1}^{\infty} \frac{k^2-x^2}{k^2}[/tex] it follows that

    [tex]\frac{y}{x}\frac{\sin \pi x}{\sin \pi y}= \prod_{k=1}^{\infty} \frac{k^2-x^2}{k^2-y^2}[/tex]

    [tex]\prod_{k=1}^{\infty} \left( 1 + \frac{2}{k^2 + 7} \right) = \prod_{k=1}^{\infty} \frac{k^2+9}{k^2 + 7} = \frac{\sqrt{7}}{3}\frac{\sin 3\pi i}{\sin \sqrt{7}\pi i}[/tex]​

    recall that [tex]\sin iz = i\mbox{sinh}z[/tex] to get the final value, namely

    [tex]\boxed{\prod_{k=1}^{\infty} \left( 1 + \frac{2}{k^2 + 7} \right) = \frac{\sqrt{7}}{3}\frac{\mbox{sinh} 3\pi}{\mbox{sinh} \sqrt{7}\pi } = \frac{\sqrt{7}}{3}\frac{e^{3\pi}-e^{-3\pi} }{e^{\sqrt{7}\pi}-e^{-\sqrt{7}\pi} }}[/tex]​
  13. Aug 24, 2006 #12


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    Wow! Are you studying math at the uni? Or how come you can solve all that?
  14. Aug 24, 2006 #13

    matt grime

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    1. Prove from first principles that exp(x) is indeed (1+x/s)^s as s tends to infinity.

    2. If M is a matrix over the complex numbers and Tr(M^r)=0 for all r show that all eigenvalues of M are zero.

    3. If f is a function from C to C and the integral of f round any triangle is zero show that f is analytic/holomorphic.

    4. What is the genus of the Riemann surface corresponding to w=sqrt((z-1)(z-2)(z-3)..(z-n))

    5. Prove, using homology groups, the fundamental theorem of algebra. Hint, this is actually a fixed point theorem.

    (Note: one of these is tricky, one of these probably requires more material than you've yet learnt.)
    Last edited: Aug 25, 2006
  15. Nov 5, 2006 #14


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    I'll do the easy one (it was homework in grad real analysis with Papa Rudin).

    I assume that we have the definition [tex]e^x=\sum_{k=0}^{\infty}\frac{x^k}{k!}[/tex].

    Let [tex]f_{s}(x)=\left(1+\frac{x}{s}\right) ^s[/tex]. Define [tex](a)_{k}:=a(a-1)\cdots (a-k+1)[/tex] and [tex](a)_{0}=1[/tex] (that is put [tex](a)_{k}= \frac{\Gamma (a+1)}{\Gamma (a-k+1)}[/tex] for [tex]k\in\mathbb{N}[/tex]. Now notice that

    [tex]f_{s}^{(k)}(x)=\frac{(s)_{k}}{s^k}\left(1+\frac{x}{s}\right) ^{s-k},[/tex] for [tex]k\in\mathbb{N}[/tex]​


    [tex]f_{s}^{(k)}(0)=\frac{(s)_{k}}{s^k}[/tex] for [tex]k\in\mathbb{N}[/tex]​

    so that we have the MacClaurin Series for [tex]f_s(x)[/tex] as being

    [tex]f_{s}(x)=\sum_{k=0}^{\infty}\frac{(s)_{k}}{k!}\left( \frac{x}{s}\right) ^{k}[/tex]​

    Consider the quanitity

    [tex]\left|f_{s}(x)-e^x\right| = \left|\sum_{k=0}^{\infty}\frac{(s)_{k}}{k!}\left( \frac{x}{s}\right) ^{k}-\sum_{k=0}^{\infty}\frac{x^k}{k!} \right|= \left|\sum_{k=0}^{\infty} \frac{x^k}{k!}\left(\frac{(s)_{k}}{s^k}-1\right)\right| [/tex]
    [tex] \leq \sum_{k=0}^{\infty} \frac{|x| ^k}{k!}\left|\frac{(s)_{k}}{s^k}-1\right|[/tex]​

    and note that [tex](s)_{k}=s(s-1)\cdots (s-k+1) \sim s^k\mbox{ as }s\rightarrow\infty[/tex] so that we have [tex]\left|f_{s}(x)-e^x\right|\rightarrow 0, \mbox{ as }s\rightarrow\infty[/tex].
  16. Nov 5, 2006 #15
    derive a variation for the nambu-goto action!
  17. Nov 5, 2006 #16


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    BTW, Hurkyl, this problem was PUTNAM 1986/A-3.

  18. Nov 5, 2006 #17


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    Oh, I didn't know that!
  19. Nov 5, 2006 #18
    How much wood could a woodchuck chuck if a woodchuck could chuck wood?

    Answer in terms of the variables x, y, and photons.

    Then the cubed root of it.

    This one's been bothering me personally.

  20. Nov 5, 2006 #19
    Ah, double posted my question.
    Last edited: Nov 5, 2006
  21. Jan 5, 2007 #20
    How about this one:


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