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:
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:
> is such that if smooth and continuous:
> So that the function :
> is a measure of the roughness.
> Putting this measure in an Lyapunov average type function:
> 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
> 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:
URL : http://home.earthlink.net/~tftn
URL : http://victorian.fortunecity.com/carmelita/435/
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