HOMEWORK #1 - part 2 : The random set
(due-Mon 3/6)

• If the function f(x)=4x(1-x) is iterated starting at x=0.3 and the results of each iteration make up a sequence, is the sequence random or deterministic?

The sequence is deterministic because you know where you start and the position in the sequence is predictable and repeatable for that orbit. If you were given the sequence, but not the orbit, it would not appear to be deterministic, but it does follow organized methods that determine the next place in the sequence.

• A function creates a binary sequence using the iterations of the sequence above giving the ith term a 0 if the number in the ith position is less than .5 and 1 if greater than .5. Is the binary sequence determinitistic or random?

The sequence is random because each iteration is equally likely to be above or below .5.

• Two sample sequences of 1's and 0's called a and b respectively. Are either of them random? If so, which one?

We looked at several dynamical systems that yeilded information about this question: the basic tent, which gave us the middle-anything set, the complex map from the circle onto the circle, and the mod function to name a few. They each told us something about the issue of determining outcomes from this type of situation. The way we defined a unitary function h(x) yeilded a 1 half the time and a 0 half the time. We calculated probabilities on this to confirm.
We also showed that each iteration is independent, so there is no overlap. What we saw was that drawing numbers at random from this sort of thing is never better (more accurate at predicting) than flipping a coin, and the probability that the numbers drawn do not have a pattern repeating is 1.

This is interesting because the deterministic dynamical system has predictable outcomes after the first random draw, but after the first n iterations, you don't know where you are within some delta of error. Which means that no matter where you are, you will not know where you are very quickly, and your accuracy (at determining where you are) decreases with each future iteration.

Out of determinism emerges randomness.

What this means for us and our assignment is that our strings both have the same property of randomness. If one was created by "flipping a coin"and the deterministic one produces the randomness of "flipping a coin" then we should not be able to tell the strings apart. Perhaps we could design a complex computer program that guesses initial conditions and checks against the
strings.

My answer is "I don't know" or better "You can't know."