Let F(x) be a continuous function of x with |slope|=2, which maps some real interval to itself, for example if |x|<1 then F(x)=2+2x for -1<x<-1/2, -2x for |x|<1/2, 2-2x for 1/2<x<1. Build an analog circuit to compute F(x) on some voltage interval. (A circuit made of ideal op-amps and ideal diodes can do that.)
--incidentally, let me pose this as a PUZZLE. Your task is to 1. invent a continuous function F(x) of this type, self-mapping the interval [-1,1]. Not necessarily the same as my example F(x). It must have |slope|>=2 everywhere. 2. Also, it should map an x that is uniform random on [-1,1] to another uniform. 3. design an analog circuit to compute F(x) using ideal op-amps, diodes, and resistors. 4. In such a way that your circuit is as simple & cheap and fast as possible. 5. Also, your F(x) should not have any "short orbits that pass thru corners." By "orbit" I mean starting from some x, if you iterate F, you after k iterations return to the starting point (that is the meaning of "length-k orbit") and by "orbit passing thru a corner" I mean, if the graph of F(x) has a corner at some value of x then the orbit of that particular x should be long (large number of iterations before pass thru either another or the same corner). This is necessary to avoid the problem that in the real world's graph of F(x) these corners will be rounded off a little, and then you could get an attracting orbit, which you do not want. You want all orbits to be repelling. This is a good optimal-design puzzle and it is not clear to me what the best possible answer is, although at one time I had a few candidates. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)