blueollie

Chicago and Nash

Workout notes: swam: 2000 straight laps (not timed; counted laps by 5: that is, 8 x 250 with no rests), then 200 pull.

I then took my daughter to Chicago Midway; we left at 9 am, I took her through the boarding pass line and to security; then I drove home and got home at 2:45. That was as fast as I’ve ever made this trip (round trip).

News: John Nash was killed in a traffic accident a couple of days ago. He was the focus of the book/movie A Beautiful Mind. He is known for the Nash equilibrium (in game theory); it was for that he won a Nobel in economics. But he had other great mathematics results; one of these was the Nash Embedding Theorem. The statement is somewhat technical but I think that I can give a flavor of what it was about.

Consider the circle. It is an object known as a “1-manifold” in that, if one examined the circle very “locally”, one would see that it was impossible to distinguish from a straight line. Example: if a tiny, near sided creature lived on a circle, it would look like a line to the creature, just like our spherical earth looks flat to us (locally).

Well, one can place a circle in a plane (distortion is allowed) in different ways:
isometric

Look at the two closed curves above. Those are “embeddings” (one to one, continuous maps from the circle). The one of the left: two points on the circle itself are “distant” from each other if and only if they are distant from each other as points on the plane. That is said to be an “isometric” embedding of the circle; points on the circle are far away from each other if and only if they are far away from each other in the plane.

Now look at the bent “circle”. See how two points of the circle are “close together” as points on the plane, but if one was forced to go from one of those points to the other point WHILE STAYING ON THE CURVE, one would have to travel a much further distance. That is a non-isometric embedding of the circle as two points are close together as points in the plane but NOT close as points on the circle.

So, the Nash embedding theorem deals with isometric embecdings; he gives a mathematical condition which guarantees that an arbitrary embedding can be approximated by an isometric embedding (as well as a dimensional criteria).

May 26, 2015 - Posted by | mathematics, swimming, travel

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