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Conference rusure::math

Title:	Mathematics at DEC

Moderator:	RUSURE::EDP

Created:	Mon Feb 03 1986
Last Modified:	Fri Jun 06 1997
Last Successful Update:	Fri Jun 06 1997
Number of topics:	2083
Total number of notes:	14613

964.0. "More geometry/analysis problems" by HPSTEK::XIA () Tue Nov 01 1988 18:33

    To Lynn, Dan, etc.
    
         I was very interested in the "real world" applications of the
    problem as stated in 960.  Here are two more theorems that are of
    similar nature.  Maybe you folks can tell me some "real world"
    applications for them too.
                                               n
    1. (The Brouwer fixed point theorem)  Let D  be the n-ball, i.e.,
        n                                n
       D  is the set of all points x in R  such that ||x|| <= 1.
               n     n
       Let f: D --> D  such that f is continuous.  Then
                                  n
       there exists a point x in D  such that:
    
                   f(x) = x
    
       I was told that this theorem has applications in economics, but
    how?
             n                                                  n
    2.  Let S  be the n-sphere (same as Sn in note 960), i.e., S 
                                       n+1
        is the set of all points x in R    such that ||x|| = 1.
    
        If n is even, then there does not exists a none zero tangent
                         n
        vector field on S .  
    
        I was told theorem has applications in meteorology.  
    
    Any details about the applications?  
    
    Another point, the above theorems only depend on the topology of
     n      n                                          n     n
    D  and S .  Meaning you don't have to have a real D  or S .
    As long as you have something that can be molded into 
     n     n
    D  or S, the theorem will work (ain't that nice).
    
    Eugene

T.R	Title	User	Personal Name	Date	Lines
964.1	some examples	AITG::DERAMO	Daniel V. {AITG,ZFC}:: D'Eramo	`Tue Nov 01 1988 20:44`	20
	For the second theorem, with n=2, you get the theorem thet you can't comb the hair on a sphere. Now, you could have a Don King sphere where the hair stands striaght out all over. But if you want to comb the hair flat [here we model a strand of hair laid flat as a vector tangent to the surface of the sphere] continuously, you can't do it. You always end up with a singularity [such as a part] or you have a point where continuity is kept by the hair length going to zero [so that the vector vanishes there]. For the first one, if you stir your coffee "continuously" then there is always a point that is at the same place in the cup. Or if you put a map of the U.S.A. over the U.S.A. and bend and stretch but not rip it, there will be a point on the map directly over the corresponding point in the country. [You could get this by contractions in a complete metric space have a fixed point, too. In all of these you have to stay within the original volume/area. Dan
964.2		CTCADM::ROTH	Lick Bush in '88	`Wed Nov 02 1988 06:33`	18
	Topological existance proofs are very useful for solving sets of nonlinear equations in terms of fixed points. An example is the recently developed method of solving low order systems of polynomials via "homotopy continuation" - it's a somewhat slow method which only works well in practice for fairly low-order systems (fewer than about 5 or 10 equations in as many variables) - but these are common in applications. There is now a set of routines called HOMPACK available for doing this. So this is a useful example. Other fascinating examples would be problems in classical mechanics, such as the theoretical long term stability of the solar system or other n-body problem. Though this is somewhat academic as it ignores relativistic dissapative effects it nonetheless leads to some surprising insights. There is a connection with ergodic theory here. - Jim
964.3		ATLAST::FRAZER	Je suis prest!	`Wed Nov 02 1988 09:11`	12
	> For the first one, if you stir your coffee "continuously" > then there is always a point that is at the same place in > the cup. Or if you put a map of the U.S.A. over the U.S.A. > and bend and stretch but not rip it, there will be a point > on the map directly over the corresponding point in the > country. [You could get this by contractions in a complete > metric space have a fixed point, too. In all of these you > have to stay within the original volume/area. If you heat your frozen dinner in the microwave for seven minutes and rotate 90� after 3 minutes and 5 minutes it will still have an icy spot or frozen spot. %^)
964.4		AITG::DERAMO	Daniel V. {AITG,ZFC}:: D'Eramo	`Wed Nov 02 1988 19:47`	11
	I just realized that "can't comb the hair on a sphere" can have a meteorological interpretation. Specify the wind velocity as a tangent vector to the earth's surface, i.e., direction and magnitude. (Ignore any vertical component.) Then the second theorem shows that somewhere on the surface of the earth the wind has zero velocity (or is not continuous). Dan [every time I tries to type "wind" it came out as "window"]