| 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 |
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 23:44 | 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
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| 964.2 | CTCADM::ROTH | Lick Bush in '88 | Wed Nov 02 1988 09:33 | 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
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| 964.3 | ATLAST::FRAZER | Je suis prest! | Wed Nov 02 1988 12:11 | 12 | |
| 964.4 | AITG::DERAMO | Daniel V. {AITG,ZFC}:: D'Eramo | Wed Nov 02 1988 22:47 | 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"]
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