Consequently, nonlocal distance (8) is indeed an intrinsic way of describing self-similar fractal, since it not only determines the dimension of a fractal curve (e.g., Cantor ternary set) but also reflects the correlation between its parts.
To study the analytic properties of a fractal curve, we define the fractal derivative (see (A.26) in Appendix A) in the form:
where [bar.f](l) = f[x(l)] is a differentiable function with respect to coordinate l, l is a parameter (e.g., the single parameter of Peano's curve [10]; for details see Appendix A) which completely determines the generation of a [omega]-dimensional fractal curve, and x(l) denotes the length of the corresponding fractal curve.
If [sub.l][D.sub.m]f(x)/[sub.l][D.sub.m]x = g(x) then the fractal integral of g(x) on a m-dimensional fractal curve [[beta].sub.m](l) is defined in the form:
where W denotes the definitional domain of the characteristic parameter l; also, the parameter l completely determines the generation of the m-dimensional fractal curve [[beta].sub.m](l).
If x(r) = [omega](m)[r.sup.m] not only describes the length of a m-dimensional fractal curve [[beta].sub.m](r) but also denotes the volume of a m-dimensional sphere (for example, the m-dimensional fractal curve [[beta].sub.m](r) fills up the entire sphere [OMEGA].) [OMEGA] and if f(x) = [bar.f](r) is a spherically symmetric function, then the fractal integration of f(x) on the m-dimensional fractal curve [[beta].sub.m](r) equals
If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (i = 1,2,3) not only describes the length of a [D.sub.i]-dimensional fractal curve [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], ([r.sub.i]) but also denotes the volume of a [D.sub.i]-dimensional sphere [[OMEGA].sub.i] and if f(r) = f([r.sub.1], [r.sub.2], [r.sub.3]) is a spherically symmetric function, then the fractal integration of f(r) on the fractal graph [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] equals
Obviously, [([DELTA]l).sup.m] is a m-dimensional Hausdorff measure, which can describe the length of a m-dimensional fractal curve. Consequently, (A.15) and (A.16) together imply that [[DELTA].sup.m]x(l) can be also thought of as a m-dimensional measure.
We call (A.19) the "nonlocal distance," which describes the length of a m-dimensional fractal curve.
Before proceeding to introduce the definition of fractal derivative, let us consider a [omega]-dimensional fractal curve [[beta].sub.[omega]]0,(1) (see Figure 3), which is determined by an independent characteristic parameter l (e.g., the fill parameter of Peano's curve), filling up a [omega]-dimensional region.
For any differentiable function y = f(x), if x = x(l) is not only a [omega]-dimensional volume but also describes the length of a [omega]-dimensional fractal curve [gamma].sub.[omega]](l), then the fractal derivative of y = f(x) with respect to the fractal curve [gamma].sub.[omega]] (l) is defined as
