By P. Ciarlet
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Extra info for An Intro. to Differential Geometry With Applns to Elasticity
The formula is chosen randomly from a pool at each step of the feedback process. In other words, the output does not just depend on the input, much like in the case of machines with memory. Unfortunately, however, there is no standard trick to rewrite the process as a (deterministic) one-step machine. If the number of formulas is N, then the wheel of fortune has N segments, one for each formula. The size of the segment can be different for each of them in order to accommodate for different probabilities in the random selection mechanism.
His book also contained the following problem, which has inspired people ever since. There is one pair of rabbits which is born at time 0. , every month a new pair is born to the original pair). Moreover, each new pair of rabbits matures after one month and begins producing pairs of offspring every month after that ad Infinitum. One assumes that the rabbits live forever. What is the number of pairs after n months? Let us be careful and follow the evolution of rabbits step by step. In our rabbit population, let us distinguish between adult and young pairs of rabbits.
Moreover, we can then rewrite the set of formulas Xn+t = g(xn, Yn), Yn+l = Xn, by a single formula: Zn+I = G(Zn). In other words, we do not have to go to the trouble of developing a special machine for two-step methods. They are perfectly covered by one-step machines. The Rabbit Problem as One-Step Machine Let us give one example, the Fibonacci numbers, defined by the two-step method with Ao = 0 and A 1 = 1. The equivalent equations for a one-step method operating on pairs (xn, Yn) is Xn+l Yn+l = Xn + Yn = Xn with initial settings x 0 = 0 and y0 = 1.