A Study of Some Scientific Computing Questions on Fractal Theory

Abstract: In this paper, we discuss some scientific computing questions on fractal theory. The paper consists of 4 chapters. Chapter 1 is an Introduction. The relationship between the convergence ball of some kinds of iterations in Banach space and their dynamical behavior on Riemann spheres areinvestigated in chapter 2. We find some kinds of iterations such as Euler iteration Enf(x), underrather extensive circumstances, the radius of whose convergence balls are determined accurately by their exclusive fixed points on Riemann sphere.The Euler iteration Ef is defined as follows:Ef(x) = x-(I-Pf(x))f'(x)-1f(x), whereWe assume that L has both non-decreasing and positive derivative L' in the interval [0, r0)and L() > 0. In particular, L may be taken to be polynomials of degree more than or equal to 1with positive coefficients or to be analytic functions with positive Maclaurin coefficients. Then we have proved the following results.Theorem 2.3.1 The Euler iteration for h has a unique fixed point rE in the interval(0, r0), which is an exclusive fixed point.Theorem 2.3.2 Suppose that f satisfies thatwhere x , . Then for all , the Euler iteration {Enf(X0)} convergence to x and satisfies thatwhereFurthermore, the radius rE of the convergence ball is optimal as a constant that only dependsupon L but not fFor Smale's condition, we can takeWe draw the graph 2.4.1 under the condition. This graph was taken as the cover graph by Chinese science bulletin vol. 47, No. 19 (2002).The result of this chapter published in [WhLW2002]. And the question of further study is the relationship between the convergence ball of the iterations of Euler-Hally family (2.5.1) with 3-rd order in Banach space and its dynamical behavior on Riemann spheres.In the chapter 3, we discus the theories and the techniques of making fractal graphs. In order to draw some graphs of Mandelbrot sets, we work out the problem of continuous tracing of algebraic curves and then prove the following theorem.Theorem 3.2.1 Assuming that d degree algebraic equation with complex parameter /have d roots as x1(t) , X2(t) , ……, xd(t), and they are all single root, then for eachi = 1, 2, ….., d , while t has increment At, the increment of xt (t) isThus if we have all xi(t) , we can solve all xi(t + Δt) by Weierstrass parallel iteration.[WH1996]The formula of this iteration isWe selectif |Δt| is small enough, then we haveWe also put forward the continuous fill arithmetic of plane. After the combination of this arithmetic and the continuous tracking arithmetic of algebraic curves, we can give the correct number of all roots of algebraic equation whose parameter picked up from a square of complex plane, thus we can draw all of the graphs of Mandelbrotsets.Furthermore, we also solve the theories and techniques problems of making Sullivan field, the technical questions of how to show the Julia sets and plenary Julia sets which draw the high quality graphs of ref. [W1998b], [WH1996], [WH2000] and [WL2001].The part of main result was published in [WhWW2000]. And the further study still reeds lots of valuable fractal graphs to be drawn.In chapter 4, we study how to search the upper estimation of the Hausdorff measure of regular fractal sets. By the analyses of Sierpinski gasket as a typical example, we provide the following set of strategy in fact:1. Choose an abundant cover sets according to the symmetry of the fractal set;2. Build the coding arithmetic by the generation rules of the pre-fractal sets in the cover sets;3. Establish coding arithmetic for the cover sets with arc border in assisting with tracing technique of theories on numerical control;4. Let the computer choose the optimization automatically.Using the above methods, we find the upper estimation of Sierpinski gasket is the best so farThe main results of this chapter were published in [Whl999] and [WhW1999]. And the question of further r…
Key words: computably enumerable degree; minimal pair.

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