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Picards theorem

  1. Oct 21, 2011 #1

    georg gill

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    I was wondering if anyone could explain the mathematic signs in the first line in detailed proof
    in the link here. What do this mathemathical sentence mean sign by sign?

    http://bildr.no/view/1002076
     
  2. jcsd
  3. Oct 22, 2011 #2

    LCKurtz

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    You mean the line where he indicates that the closed unit balls in this case are just closed intervals?
     
  4. Oct 23, 2011 #3

    georg gill

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    The mathematical signs in dotted area here between: Let ........... be the compact cylinder where f is defined this is


    Which is just below the header detailed proof. It looks like the sign for cross product to me but how does that make a cylinder?
     
  5. Oct 23, 2011 #4

    LCKurtz

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    That is the symbol for the Cartesian Product of the two sets. The Cartesian product of A and B is:

    A x B = {(a,b): a ε A and b ε B}
     
  6. Oct 28, 2011 #5

    georg gill

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    I have read some about the cartesian product with a deck of cards as example which has 13 different cardvalues and 4 different colors which make a deck of cards have cartesian product equal 52.

    But how can a cartesian product descripe a cylinder?
     
  7. Oct 28, 2011 #6

    LCKurtz

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    It is using cylinder in a more general sense than a common circular cylinder. If you take a circle in the xy plane and take its Cartesian product with the z axis you get a what anyone would call a cylinder. But you can take any region, such as a square in the xy plane and cross it with the z axis. You get an infinitely long square cross section block. Just as you would call the surface of that block a cylindrical surface, you would also call the block itself a cylinder. It just isn't round.
     
  8. Nov 9, 2011 #7

    georg gill

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    This is the whole proof

    http://en.wikipedia.org/wiki/Picard–Lindelöf_theorem#Detailed_proof

    I wonder is t who is the variable for I a parameter for two dimensions (thoose two dimensions one could call that x and y-axis?) where as y is a variable for B which makes the third dimensions (one could call that one z-axis)

    and does the points of I make a circle and B make a line on the z-axis to make the points on the surface of a cylinder?

    And what does M=sup||f|| mean?
     
  9. Nov 9, 2011 #8

    LCKurtz

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    Think of a t-y plane instead of xy plane. You are looking for a solution of the DE with y(t0)= y0. Ia is just the closure of the open interval of length 2a about t0: [itex]I_a=\overline{(t_0-a,t_0+a)}= [t_0-a,t_0+a][/itex].

    Simarly, Bb is the closure of an interval of length 2b about y0 on the y axis: [itex]B_b=\overline{(y_0-b,y_0+b)}= [y_0-b,y_0+b][/itex]. Your picture looks like this:

    picard.jpg

    Your Cartesian product in this case is just a rectangle in the ty plane. What is confusing you is that the author is writing it in a more general notation to use the general Banach Fixed Point Theorem.

    For continuous functions and a closed region, the sup of a function is the same thing as its maximum.
     
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