Re: A functional measure of roughness

*To*: mathgroup at smc.vnet.net*Subject*: [mg50252] Re: A functional measure of roughness*From*: "Roger L. Bagula" <rlbtftn at netscape.net>*Date*: Mon, 23 Aug 2004 06:34:14 -0400 (EDT)*Organization*: bmftg/tftn*Reply-to*: tftn at earthlink.net*Sender*: owner-wri-mathgroup at wolfram.com

The measure of roughness based on the 3 point Bezier which is like a Lyapunov exponent average, but more sensative: It turns out to be like a second derivative: Measure[n]=Sum[Log[1+Abs[f''(x(i))]/4],{i,1,n}]/n Limit[Measure[n],n-> Infinity]=rho The roughness measure is relate to the Lyapunov exponent average by k=Sum[Log[f'(x(i))],{i,1,n}]/n/Sum[Log[1+Abs[f''(x(i))]/4],{i,1,n}]/n k is close to 2 for the primes. So I have actually found two measures. Roger Bagula wrote: > In thinking of a way to get a better than Lyapunov , Hausdorff or Kolmogorov > measure of dimension , I thought of this: > F(curve)=0 if smooth and continuous > F(curve)<>0 if rough or discontinuous > The best measure of dimensional roughness (Mandelbrot's way of > expressing it) is the > Lyapunov exponent (or maybe the Hurst exponent?). > Box counting or capacity/ entropy dimension of the Kolmogorov type > is too big most of the time > while Hausdorff being very cut-off measure like > is usually too small. > The trouble with Lyapunov is that it depends on a derivative > and unless you are talking about a fractional derivative, > many fractal functions are of the Weierstrass fractal type > where the classical derivative doesn't exist. > > I did some work on Bezier functions in IFS in the past > and fractional partial derivatives of an angular sort as well. > I came to realize that the three point Bezier function of an iterative > sequence in n: > Bezier[p,n]=p^2*f(n+2+2*p*(1-p)*f(n+1)+(1-p)^2*f(n) > is such that if smooth and continuous: > f(n+1)=Bezier[1/2,n]=f(n+2)/4+f(n+1)/2+f(n)/4 > So that the function : > delta[n]=f(n+2)/4+f(n+1)/2+f(n)/4-f(n+1) > is a measure of the roughness. > Putting this measure in an Lyapunov average type function: > Measure[n]=Sum[Log[1+delta[i]],{i,1,n}]/n > I tried this out by comparing it to a known rough set, the primes > and it's Lyapunov integer difference average. > In this experiment the new Bezier roughness measure performs better than the > Lyapunov equivalent over the same range in detecting roughness. > Respectfully, Roger L. Bagula > > tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: > 619-5610814 : > URL : http://home.earthlink.net/~tftn > URL : http://victorian.fortunecity.com/carmelita/435/ > > > Respectfully, Roger L. Bagula tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : URL : http://home.earthlink.net/~tftn URL : http://victorian.fortunecity.com/carmelita/435/