The superfractal of V -variable fractal measures in .NET Maker Code 39 Extended in .NET The superfractal of V -variable fractal measures

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5.17 The superfractal of V -variable fractal measures using barcode maker for visual .net control to generate, create code 3 of 9 image in visual .net applications. USPS Confirm Service Barcode Just as in Section 5.11 w Visual Studio .NET 39 barcode e de ned the 1-variable measure IFS F P(1) similarly to the way in which we de ned the 1-variable IFS F (1) , so here we de ne the V -variable measure IFS F P(V ) similarly to the way in which we de ned F (V ) .

The key difference is that now we work in the space PV = P(X)V instead of the space H(X)V. As elsewhere, we tie all metrics back to the metric spaces (X, dX ), (H(X), dH(X) ) and (P(X), dP(X) ), in the manner described in Section 1.13.

Here we merely point out the form of the V-variable measure IFS. Let V N, let A be the index set introduced in Equation (5.12.

1), let the superIFS {X; F1 , F2 , . . .

, F M ; P1 , P2 , . . .

, PM } be as above and let probabilities {P a . a A} be given as in Equ ation (5.12.2); here we use P a in place of f a .

Then we de ne, in the manner which you might already have guessed, f a : P(X)V P(X)V by f a ( ) =. L m1 l=1 L m2 l=1 LmV l=1 plm 1 flm 1 v1,l ,. plm 2 flm 2 v2,l , . . .

,. plm V flm V vV,l (5.17.1).

for all = ( 1 , 2 , . .net framework bar code 39 .

. , V ) P(X)V . We de ne the V -variable measure IFS F P(V ) to be F P(V ) := P(X)V ; f a , P a , a A .

T h e o r e m 5.17.1 For V = 1, 2, .

. . , F P(V ) is a hyperbolic IFS.


Superfractals Figure 5.35 Three success visual .net 39 barcode ive fractal measures belonging to a 2-variable superfractal.

The pixels in the support of each 2-variable measure are coloured either black or a shade of green. The intensity of the green of a pixel is a monotonic increasing function of the measure of the pixel..

Proof See [16].. Everything works analogou sly to the case of F (V ) but now the underlying space consists of measures instead of sets. The set attractor A P(V ) of F P(V ) is a set of measures in P(X)V. The set of components of the elements of A P(V ) is a superfractal, which we may denote by A P(V ) .

It consists of V -variable fractal measures. The elements of A P(V ) are distributed on P(X)V according to the probability measure P(V ) P(P(X )V ), which is the measure attractor of F P(V ) . The measure P(V ) de nes a marginal probability distribution P(V ) , obtained by projecting it onto a single component, and this measure describes the asymptotic distribution of measures obtained, almost always, by following chaos-game orbits for F P(V ) and keeping only the rst components.

In Figure 5.35 we show some examples of 2-variable fractal measures, rendered in shades of green according to pixel mass. This example corresponds to the same superIFS as that used in Figure 5.

24. The probabilities of the functions in the IFSs 1 2 1 2 are p1 = p1 = 0.74 and p2 = p2 = 0.

26. The IFSs are assigned probabilities P1 = P2 = 0.5.

I think that by now you will have got the idea. There are many fascinating kinds of V -variable objects that may be de ned and explored both mathematically and experimentally. If you do this in the context of either scienti c or engineering applications, rich rewards may be obtained.

This is new territory!.
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