How about some mathematics eh? Fixed Points

. Everybody is familiar with the idea of a mathematical formula. A formula looks like just a bunch of incomprehensible (or worse: comprehensible) symbols with unknowns, like x and y and so on. You put in actual values in for x and y etc, and bingo, you get a number!! The formula is simply a converter that converts values of the variables into a number. The idea of a function is very similar: it is something that converts particular values of variables into a number. The simplest case is where there is only one variable, say x. Values of x are converted into numbers, usually called y values. Some functions can only use x values in a specific set. This set of allowed x values or inputs is called the domain. (Think of a coin operated machine that only accepts certain coins.) The entire collection of possible numbers that the function gives out, in other words all the possible outputs, is called the range of the function. So the domain is the collection of all possible inputs, and the range is the collection of all possible outputs. What if the range is actually a subset of domain? For example, what if all the incoming x values lie between 0 and 10, and the function gives outputs between, say, 2 and 7? Then there is a possibility that the output might be exactly the same as the input for one or more x values! Consider the function f(x) = 3x^3 - 2x, shown at right. (We're using the convention that x^3 represents x times x times x, or raised to the power 3.) When x is any one of 0, 1 or -1, f(x) = x. Those three values are called the fixed points of the function. The Brouwer Fixed-Point Theorem says that if f is a continuous function from the interval [0,1] into the interval [0,1], then f must have at least one fixed point! To explain: the interval [0,1] simply means all the numbers between 0 and 1 inclusive. Continuous is a little harder; it means that the function f must only vary so that small changes in x result in small changes in the output. In particular, as x varies gradually from 0 to 1, f is not allowed to take jumps in its output values. The Contraction Mapping Theorem says that if there is a positive number less than 1, say k, such that the distance between outputs from two values u and v are less than k times the distance between u and v, then the function has a unique fixed point.