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Re: A functional measure of roughness

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

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:
Limit[Measure[n],n-> Infinity]=rho
The roughness measure is relate to the Lyapunov exponent average by
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 
 > 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, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel:
 > 619-5610814 :
 > URL :
 > URL :

Respectfully, Roger L. Bagula
tftn at, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 
619-5610814 :

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