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  1. Jul 1, 2007 #1
  2. jcsd
  3. Jul 14, 2007 #2
    Consider the following space [itex]X[/itex], consisting of two [itex]2[/itex]-spheres and two arcs glued together. Compute its fundamental group.

    Since I can't draw a picture online, call the first sphere [itex]S_1[/itex] and call the second sphere [itex]S_2[/itex]. Then one arc connects [itex]x_1 \in S_1[/itex] to [itex]x_2 \in S_2[/itex] and another arc connects [itex]y_1\in S_1[/itex] to [itex]y_2 \in S_2[/itex], where all the points are distinct.

    I thought about this problem and contracted the arcs (so it looks like two 2-spheres identified in two points), and I want to use van Kampen. But I'm having a hard time figuring out two open sets [itex]A[/itex] and [itex]B[/itex] so that [itex]A \cup B = X[/itex] and [itex]A\cap B[/itex] is path connected.

    Thank you!

    Actually, can I contract one of the arcs so that the two 2-spheres touch at one point, then move the points [itex]x_1, x_2, y_1, y_2[/itex] to the point of intersection? Then I have [itex]S^2 \vee S^1\vee S^2[/itex]. Then [itex]\Pi_1(X) = \mathbb{Z}[\itex]. Is this a correct analysis?
     
    Last edited: Jul 14, 2007
  4. Aug 9, 2007 #3

    Astronuc

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    Staff: Mentor

    This subject seems to be coming up quite often, especially for those studying wave mechanics.

    Here is a brief intro. I'll be looking for better articles and I hope others will contribute references or insight from personal experience.

    http://en.wikipedia.org/wiki/Dispersion_relation

    and related topics
    http://en.wikipedia.org/wiki/Group_velocity
    http://en.wikipedia.org/wiki/Phase_velocity

    http://tosio.math.toronto.edu/wiki/index.php/Dispersion_relation - note that this is a wiki page from Department of Mathematics at U. Toronto

    I also hope to address anharmonic and non-linear systems.

    I think it important for students to understand the significance of the higher order spatial (and temporal) derivates with respect to the dependent variable, uxxx or [itex]\partial^3_x\phi[/itex] in the KdV equation.
     
    Last edited: Aug 9, 2007
  5. Aug 14, 2007 #4
    I was looking for a reference for something relating to something mentioned, and ran across a nice site for a lot of references. If there is a thread that it can be re-posted--let me know.

    http://web.mit.edu/redingtn/www/netadv/


    This site organizes topics from various sources including arXiv alphabetically for you.

    ---------------------------------------------
    oh--and I still couldn't find what I was looking for though
     
    Last edited: Aug 14, 2007
  6. Sep 24, 2007 #5

    EnumaElish

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    Homework Helper

  7. Sep 24, 2007 #6
    well i dont know how much you can trust a site where it says there something like:
    "physics hypothesis that there is no speed barrier in the universe"

    i would say quite unpopular approach is it not?
     
  8. Sep 28, 2007 #7
    Agreed 'loop"...and if you nav to the home page it is filled with java errors that make looking frustrating and counterproductive.
     
  9. Jan 27, 2008 #8
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