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Acoustic communication impossible in even-D

  1. Apr 24, 2013 #1

    atyy

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    Arnold, an in interview with Lui in Notices of the AMS, Volume 44, no 4, p432 says:

    "It is a far-reaching generalization of the well-known fact of the impossibility of acoustic communication in the even-dimensional spaces (for instance, in the “plane” world), while in our three-dimensional world we communicate easily."

    What is he referring to?
     
  2. jcsd
  3. Apr 24, 2013 #2

    Danger

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    My first guess would be LSD.
     
  4. Apr 24, 2013 #3

    atyy

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    :confused: I can't tell if you are pulling my leg!
     
  5. Apr 24, 2013 #4

    Danger

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    Hey, now... it's me. Of course I am. (Jeez, do I have to put smilies on everything?)
     
  6. Apr 24, 2013 #5

    atyy

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    Well, I just wanted to make sure I hadn't missed a brilliant insight about Lorentz Specific Dimensions ...

    Anyway, here's more context - it has something to do with lacunas of hyperbolic PDEs.

    "Petrovskii was no longer active in mathematics. However, he was extremely important for the Moscow mathematical community, always trying to support genuine mathematicians in difficult fights with the Communist Party.

    His mathematical taste was rather classical, based on the Italian school of algebraic geometry more than the set-theoretic conceptions. Sir Michael Atiyah once told me that he was always delighted by the way Petrovskii dealt with algebraic geometry in his works on PDEs. One of these, the paper on the lacunas of hyperbolic PDEs, was later rewritten by Atiyah, Bott, and Gårding in modern terminology in two long papers in Acta Mathematica. It is a far-reaching generalization of the well-known fact of the impossibility of acoustic communication in the even-dimensional spaces (for instance, in the “plane” world), while in our three-dimensional world we communicate easily. It is interesting that in this paper, Petrovskii proved that the cohomology classes of the complement of an algebraic variety are representable by rational differential forms—a result which is usually attributed to Grothendieck."
     
    Last edited: Apr 24, 2013
  7. Apr 24, 2013 #6

    atyy

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    Looks like Wikipedia has an article and links about this: http://en.wikipedia.org/wiki/Petrovsky_lacuna

    The Petrovsky article linked to says:

    "This circumstance implies the fact that in case p is odd the spherical wave produced in a small neighbourhood of a point Q ... has the property that both its front and back edges are sharp. As to the case where p is even or p = 1, only the front edge of such a wave is sharp, while the back edge is diffuse. In the first case it is said that there is no diffusion of waves ... in the second case — that the diffusion of waves takes place."
     
  8. Apr 24, 2013 #7

    Danger

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    That's the last thing that you need to worry about when you see a post from me.
    As most folks here know, I have a grade 9 math level. Nothing that you wrote makes any sense to me. (I can figure out the volume of a cylinder, but that's about the extent of my numerical talents.)
     
  9. Apr 24, 2013 #8

    atyy

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    How about the volume of a cone?
     
  10. Apr 24, 2013 #9

    Danger

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    :redface:
     
  11. Apr 25, 2013 #10

    atyy

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    Here's another description, maybe more readable.

    http://web.eecs.umich.edu/~gessl/georg_papers/FA05-DrumSim.pdf
    "Wakes are the content of a wave field after the point of first arrival, the wavefront, has passed. ...

    The study of the existence and basic properties of wakes goes back to Petrovsky and has somewhat later been deepened by Atiyah, Bott and Gårding [2, 3]. If a wavefront does not create a wake, they call it a lacuna. It is known since Volterra, that the wave equation in even spatial dimensions creates wakes, whereas in odd spatial dimensions greater or equal three it doesn’t. The one dimensional wave equation constitutes a special case, as a step function is the correct response to velocity excitations hence there is a “wake-like” influence after the impulsive propagation"
     
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