Fractal

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Related to Fractal curve: fractal dimensionality

Fractal

An object in which the parts are in some way related to the whole. That is, the individual components are "self-similar." An example is the branching network in a tree. While each branch, and each successive smaller branching is different, they are qualitatively similar to the structure of the whole tree.

Fractal

1. In technical analysis, an indicator of the reversal of the previous trend. It is shown on a candlestick chart as a series of five candles, representing five trading days. A bullish fractal occurs when the lowest low of any trading day is represented by the middle candle, with two successively less low trading days on each side. This is seen as a buy signal. A bearish fractal occurs when the highest high of the five days is represented by the middle candle, with two successively less high trading days on each side. This is seen as a sell signal.

2. Any whole made up of parts that are self-similar.
References in periodicals archive ?
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
In this paper, a novel CSRR using Koch fractal curve is applied to bandpass filter.