chaos fractals and julia sets

... Contents Chapter Julia and Mandelbrot Sets Constructed Using the Escape - Time Algorithm and Boundary Scanning Method 229 7.1 Julia Sets and Filled Julia Sets 230 7.2 Parameter Sets 267 7.3 Illustrations ... certain Julia sets, the Mandelbrot set and other parameter sets, using the ideas and constructions of Chapter The main ideas needed for the construction of the (filled) Julia sets of polynomials and ... Mathematica Fractals, Julia Sets, Patterns and Natural Forms This page is intentionally left blank Graphics with Mathematica Fractals, Julia Sets, Patterns and Natural Forms by Chonat Getz and Janet...

Ngày tải lên: 06/07/2014, 15:16

Ngày tải lên: 08/05/2014, 12:51

... dimension of Julia sets and similarity between the Mandelbrot set and Julia sets, Fund Math 170 (2001) 287–317 L Carleson, P Jones, and J -C Yoccoz, Julia and John, Bol soc Brasil Mat 25 (1994) 1–30 ... to the referee and the editor for reading the first version of this paper, and for very useful suggestions and comments They help me to improve this work Continuity of Julia Sets and Hausdorff Dimension ... dimension of the boundary of the Mandelbrot set and Julia sets, Ann Math 147 (1998) 225–267 M Urbanski, Rational functions with no recurrent critical points, Ergod Th ´ and Dynam Sys 14 (1994) 391–414...

Ngày tải lên: 06/08/2014, 05:20

... PURPOSES ONLY 26 Module 6: Creating and Manipulating Sets Head and Tail Functions Topic Objective To describe Head and Tail functions ! Head Function $ Lead-in The Head and Tail functions extract a ... PURPOSES ONLY Module 6: Creating and Manipulating Sets 31 Exercise Using Functions to Manipulate Sets In this exercise, you will manipulate sets to create new sets Delivery Tip The procedures ... three ways to combine two sets: the Union function, the plus operator (+), and simply enclosing the sets in braces The plus operator requires an extra set of braces anyway, and there is no difference...

Ngày tải lên: 04/11/2013, 13:15

... Chapter Root Finding and Nonlinear Sets of Equations for (i=1;i

Ngày tải lên: 07/11/2013, 19:15

... trade@cup.cam.ac.uk (outside North America) a x2 x3 b x1 d c f e 351 9.1 Bracketing and Bisection 352 Chapter Root Finding and Nonlinear Sets of Equations #include #define FACTOR 1.6 #define NTRY 50 ... float *x2) Given a function func and an initial guessed range x1 to x2, the routine expands the range geometrically until a root is bracketed by the returned values x1 and x2 (in which case zbrac ... subdivide the interval into n equally spaced segments, and search for zero crossings of the function nb is input as the maximum number of roots sought, and is reset to the number of bracketing pairs...

Ngày tải lên: 15/12/2013, 04:15

... Method, False Position Method, and Ridders’ Method f(x) 356 Chapter Root Finding and Nonlinear Sets of Equations f (x) Figure 9.2.3 Example where both the secant and false position methods will ... both reliability and speed, Ridders’ method is generally competitive with the more highly developed and better established (but more complicated) method of Van Wijngaarden, Dekker, and Brent, which ... REFERENCES AND FURTHER READING: Ralston, A., and Rabinowitz, P 1978, A First Course in Numerical Analysis, 2nd ed (New York: McGraw-Hill), §8.3 Ostrowski, A.M 1966, Solutions of Equations and Systems...

Ngày tải lên: 15/12/2013, 04:15

... problem by maintaining brackets on the root and checking where the interpolation would land before carrying out the division When the correction P/Q would not land within the bounds, or when the bounds ... Finding and Nonlinear Sets of Equations } a=b; fa=fb; if (fabs(d) > tol1) b += d; else b += SIGN(tol1,xm); fb=(*func)(b); Move last best guess to a Evaluate new trial root } CITED REFERENCES AND ... supposed to, and, if it is not, we can intersperse bisection steps so as to guarantee at least linear convergence This kind of super-strategy requires attention to bookkeeping detail, and also careful...

Ngày tải lên: 15/12/2013, 04:15

... Newton-Raphson and Fractals An interesting sidelight to our repeated warnings about Newton-Raphson’s unpredictable global convergence properties — its very rapid local convergence notwithstanding — ... North America) 363 9.4 Newton-Raphson Method Using Derivative f(x) 364 Chapter Root Finding and Nonlinear Sets of Equations f(x) Figure 9.4.3 Unfortunate case where Newton’s method enters a nonconvergent ... function value as fn and the derivative as df We have included input bounds on the root simply to be consistent with previous root-finding routines: Newton does not adjust bounds, and works only on...

Ngày tải lên: 15/12/2013, 04:15

... (outside North America) Given B and C, R and S can be readily found, by polynomial division (§5.3) We can consider R and S to be adjustable functions of B and C, and they will be zero if the quadratic ... coefficients, i=0 and the coefficients a[0 m] The method is to construct an upper Hessenberg matrix whose eigenvalues are the desired roots, and then use the routines balanc and hqr The real and imaginary ... 370 Chapter Root Finding and Nonlinear Sets of Equations f (x) f (x) (a) x (b) Figure 9.5.1 (a) Linear, quadratic, and cubic behavior at the roots of polynomials Only...

Ngày tải lên: 15/12/2013, 04:15

... 380 Chapter Root Finding and Nonlinear Sets of Equations no root here! two roots here f pos g pos M f=0 g pos g= g neg f pos f neg ... let x denote the entire vector of values xi and F denote the entire vector of functions Fi In the neighborhood of x, each of the functions Fi can be expanded Sample page from NUMERICAL RECIPES ... equation (9.6.3) is F(x + δx) = F(x) + J · δx + O(δx2 ) (9.6.5) By neglecting terms of order δx2 and higher and by setting F(x + δx) = 0, we obtain a set of linear equations for the corrections δx that...

Ngày tải lên: 15/12/2013, 04:15

... 384 Chapter Root Finding and Nonlinear Sets of Equations Line Searches and Backtracking When we are not close enough to the minimum of f , taking ... (*func)(float [])) Given an n-dimensional point xold[1 n], the value of the function and gradient there, fold and g[1 n], and a direction p[1 n], finds a new point x[1 n] along the direction p from xold ... Chapter Root Finding and Nonlinear Sets of Equations for (den=0.0,i=1;i

Ngày tải lên: 15/12/2013, 04:15

... the two following properties: (M1) If X and Y are G -sets, let ix and iv be the respective injections from X and Y into X [I Y, then the maps M*(ix) | M*(iv) and M ( i x ) | M.(iv) are m u t u a ... let Y and Y' be G -sets, and i and i' be the respective injections from Y and Y' into Y II Y' If X is a G-set, then (M~,x|174 = ( M.(Idx~)| " " '))(o M (Idxz)@M (Idxz '1) = Id and it it also ... values in A - M o d , defined for G -sets X and Y, and for a map f of G -sets from X to Y, by F(X) = F(Mx) 1.3 Construction F*(f) = F(MJx) of 7~(M,N) F.(f) = F(Mj,x) and M@N From now on, the ring R...

Ngày tải lên: 14/03/2014, 21:20

... dynamics which we understand poorly As soon as a phenomenon is understood better, it is reclaimed and renamed: “a route to chaos , “spatiotemporal chaos , and so on In ChaosBook we shall develop ... Korsch and Jodl [1.10] and Nusse and Yorke [1.11] also emphasize hands-on approach to dynamics With this, and a graduate level-exposure to statistical mechanics, partial differential equations and ... another sphere in free space and if their sizes and their paths and directions before collision are known, we can then foretell and calculate how they will rebound and what course they will take...

Ngày tải lên: 17/03/2014, 14:29

... and Expanded, edited by Stuart S.Nagel 54 Handbook of Regulation and Administrative Law, edited by David ti Rosenbloom and Richard D Schwartz 55 Handbook of Bureaucracy, edited by Ali Farazmand ... Public Productivity Handbook, Second Edition, Revised and Expanded, edited by Marc Holzer and Seok-Hwan Lee Handbook of Gay, Lesbian, Bisexual, and Transgender Administration and Policy, edited ... 40 Organizational Behavior and Public Management: Second Edition, Revised and Expanded, Michael L Vasu, Debra W Stewart, and G David Garson 41 Handbook of Comparative and Development Public Administration,...

Ngày tải lên: 29/03/2014, 05:20

... the picture and the two perspectives of the main characters depict the intricate relationship of order and chaos: chaos coming out of order, order coming out of chaos, order and chaos inherent ... CHAOS the New Book on Nature: Physics and Metaphysics in the Modern Novel, Susan Strehle in Fiction in the Quantum Universe, and N Katherine Hayles in The Cosmic Web, Chaos Bound, and Chaos and ... purposes, and/ or embrace current rhetoric of chaos theory (including tropes of order and chaos) Many different forms of fiction can profit from a consideration of, and comparison with, chaos theory, and...

Ngày tải lên: 11/06/2014, 12:51

... Narrow Banding 3.7 Merits, Demerits, Conclusions and the Future of 2-D and 3-D Level Sets in Medical Imagery 3.7.1 Advantages of Level Sets 3.7.2 Disadvantages of Level Sets 3.7.3 Conclusions and ... hearts and have shown lots of love and care for me Swamy Laxminarayan would like to express his loving acknowledgements to his wife xv xvi PDE and Level Sets Marijke and to his kids, Malini and ... Osher and Sethian [25] Equation (1.55) for 2-D and 3-D cases can be generalized as: and respectively, where and are curvature dependent speed functions in 2-D and 3-D, respectively where and are...

Ngày tải lên: 27/06/2014, 10:20

Ngày tải lên: 05/08/2014, 15:20

... ∧ a2 = a3 ∧ a4 and a1 ∨ a2 = a3 ∨ a4 , a1 ∧ a2 = a1 ∧ a3 and a1 ∨ a2 = a1 ∨ a3 (2.2) (2.3) Now, we introduce the entropy function and show an inequality of it Given a discrete random variable ... n-dimensional random variable, then n H(ξ) ≤ i=1 H(ξi ) (2.6) Recovering Set on Boolean lattice Bn In this section, we study recovering sets on Boolean lattices where we use ∩ and ∪ instead of ∧ and ∨ ... than the cardilattice, and we give a bound nalities of a general recovering pair discussed in [1], [9], and [10] Proof Let us define a random variable ξ = ∩ aj , where and aj are independently...

Ngày tải lên: 08/08/2014, 14:23