Review for Week of Apr 10

RANDOM:
The dynamical system using mod(1) was completely deterministic. We put in the string of 1's and 0's and it gives them back to us in a modified form one by one... Yet as we defined the system, all points veer to infinity.

• All the points are unstable... deterministic system yeilds chaos

This happens with our question from the homework assignment about the random string. Our defined map can be redefined to stretch out the map. We showed that the streched system mapped to itself has the same orbit structure as the original map.

• Meaning it commutes even though it is not one-to-one (i.e. yeilds chaos.)

Overall, what we observed is that many (all?) deterministic systems have inherent instability (attibuted to a measure of error accumulating with each iteration?)

• Therefore, nature is not the source of randomness. Chaos is the result of deterministic systems.

Questions:

If we had a perfect model for the universe that incorporates chaos and instability, would we still find behavior that is "random?"

Did we ever really get to what is random?

I need to ammend my statement that there are no movies in which mathematicians are featured in roles meaningful to the plot and are not crazy. I thought of a few... Jurassic Park, Sneakers (the guy who built the box) and About Schmidt (he's having a mid-life crisis, but he isn't actually crazy - but he's not a theorical mathematician either.)

GAUSSIAN DISTRIBUTION

• The sum of n independent normals with parameters (0, 1) is also normal
• The sum has parameters: E(X) = 0, V(X) = n

The drunkard walk is a sum of N in which we found that

• The variance of independent sums is independent of size of step or time it takes to take them.
• Disjoint increments (the steps) are independent of each other: The memoryless property.
• There exists a limiting process as the variance of the steps goes to 0.
• The V(X)=V(X).
• There exists an infinite number of steps that are all independent.

I don't know what "gaussian noise" is.

I will write more about this when we have talked about it further.