User:Adam majewski

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  "The answer is no, although it doesn't seem so easy to give a rigorous counter-example."  Glougloubarbaki[1]


" Mathematics takes place at different time-scales. If you can solve a problem in 55 minutes that others need an hour to solve, you can probably get a good job. If you can solve a problem in a month that others might need a year to solve, you will probably do well as a graduate student. But if you can solve a problem in 10 years that nobody else can solve in a lifetime, you could be a great mathematician." Robert Israel


  " anyone who wishes to study this topic should earn a Ph.D. in number theory and spend several years researching the relevant topics in depth, with the guidance of a world class expert." Alon Amit, PhD in Mathematics; Mathcircler.


  "The author apologises wholeheartedly to those who dare read the source code." Freddie R. Exall
  A BELIEF IS NOT A PROOF.
  "Category theory is ... the most, abstract fields of mathematics" Robb Seaton[2]
  For the sake of completeness, here is the entire code I used:"
   "That's quite a comprehensive analysis: It'll take me probably a week (or better a month) to understand it. " marcm200[3]
    I feel this is a very basic question, but I seem to be unable to find an answer for (neither by myself nor searching)  marcm200

My home page - dead (:-(

Wikipedia-logo.png Wikipedia - Adam majewski

Commons-logo.svg Commons - Adam majewski

My Dropbox public folders

Mathematics Stack Exchange

gitlab repo

Links :

2D normalized homogenous coordinate

Coordinate

  • finite point on the complex plane with Cartesian coordinate
  • point r on the extended complex plane (= Riemann sphere )
    • 2D homogenous coordinate
    • normalized 2D homogenous coordinate


Special points

  • origin
  • point at infinity
    • in homogenous coordinate : when last coordinate u is 1 then it is a point at infinity
    • in normalized homogenous coordinate


Conversions

from homogenous to Cartesian coordinate

The original Cartesian coordinates are recovered by dividing the first two positions by the third. Thus unlike Cartesian coordinates, a single point can be represented by infinitely many homogeneous coordinates. Note that here not u ( typical) but 1-u is used

The mapping from the sphere to the finite plane is

If

 
 
 

If then z is point at infinity

PBURF

 git clone https://github.com/luisjavierhernandez/PBURF.jl.git
 cd PBURF.jl
 cd src
 
julia

              _
   _       _ _(_)_     |  Documentation: https://docs.julialang.org
  (_)     | (_) (_)    |
   _ _   _| |_  __ _   |  Type "?" for help, "]?" for Pkg help.
  | | | | | | |/ _` |  |
  | | |_| | | | (_| |  |  Version 1.5.3
 _/ |\__'_|_|_|\__'_|  |  Ubuntu ⛬  julia/1.5.3+dfsg-3
|__/                   |


julia> pwd()
"/home/a/PBURF.jl/src"


julia> readdir()
1-element Array{String,1}:
 "PBURF.jl"

julia> include("PBURF.jl")

julia> import Pkg; Pkg.add("Polynomials")
julia>import Pkg; Pkg.add("Polynomials")
julia>import Pkg; Pkg.add("Colors")

function calculators

critical point

https://math.stackexchange.com/questions/502750/when-is-infty-a-critical-point-of-a-rational-function-on-the-sphere


Just choose an appropriate pair of charts. Write , where and are polynomial functions with no common zeros. We may assume is not a constant function – the constant case is trivial. There are a few cases:

first case

  • – look at the function . The derivative is given by

but it's not immediate how to extract anything useful from this expression. Instead, suppose the leading term of is while the leading term of is . We must have if (in terms of ) and the denominator is of degree .


Now there are two subcases:

  • – then the value of the expression at is ; in particular, is not a critical point of .
  • – then the value of the expression at is , so is a critical point of .


second case

  

look at the function

 . 
 

The derivative is given by

 

but again it's not clear what we can say from this.


Let be the leading term of and let be the leading term of .

We must have if , and the numerator is of degree ) or (if ) and the denominator is of degree .

Now there are four subcases:

  • – then is a Möbius transformation and the value of the expression is non-zero; in particular, is not a critical point of .
  • – if the numerator is of degree as well, then the value of the expression is non-zero; if the numerator is of degree , then the value of the expression is zero. (Both sub-subcases are possible, of course.)
  • – then the value of the expression at is ; in particular, is not a critical point of .
  • – then the value of the expression at is , so is a critical point of .

At any rate, the point is that there is no _easy_ criterion purely in terms of the values of and .

rigorous studies of nonlinear systems

  • computing enclosures of trajectories
  • finding and proving the existence of symbolic dynamics
  • obtaining rigorous bounds for the topological entropy
  • methods for finding accurate enclosures of chaotic attractor
  • interval operators for proving the existence of fixed points and periodic orbits
  • methods for finding all short cycles

https://rd.springer.com/chapter/10.1007/978-3-540-95972-4_2

methods

graphic software


text color

some notes

Some notes from | wikipedia


  • ,

  •  


A Gentle Introduction to the Art of Mathematics by Joe Fields,

parabolic/hyperbolic/elliptic

The meaning of the terms "elliptic, hyperbolic, parabolic" in different disciplines in mathematics[4]


hyperbolic

curves

https://www.mathcurve.com/courbes2d.gb/rosace/rosace.shtml rose curve = n-folium: The curve is composed of a n base patterns. The pattern is called : the petal or branch / leaf / lobe - symmetrical about Ox obtained for angle between -pi/(2n) and pi/(2n)


osculating circle of a sufficiently smooth plane curve


curvature

Interesting curves involning the curvature concept by Xah Lee:

  • Evolute curve (the centers of osculating circles)
  • Radial curve (locus of osculating circle normals)
  • circle = curve with constant curvature everywhere
  • line = curve with curvature of 0 everywhere)
  • Clothoid = spiral cirve of linearly increasing curvature)

topology

number theory


algorithms

simplify curve

geometry

Moore-Neighbor Tracing


see also:

chain code

test

( land on the root point of period 267 component : c267 = 0.250137369683480-0.000003221184145 i with angled internal adress :


( land on the root point of period 268 component c268 = 0.250137369683480-0.000003221184145i period = 10000 i with angled internal adress :

mandelbrot set

the Mandelbrot set for the function 1/z - z∙(1 + 0.001∙z)/(1 - 0.002∙z + 0.001∙z2) = "1 -0.002 - 0.999 -0.001 0 1 -0.002 0.001":

 

http://www.juliasets.dk/UFP.htm

video

programs

gradient line of the 2D scalar field

Key words:

  • "gradient line" 2d "scalar field"

flow :

  • level curves and gradient vector
  • flow across continuously-spaced level curves
  • The flow’s derivative is the gradient – the flow will follow the gradient vectors
  • gradient is the direction of steepest ascent in the zz-direction, the reverse of the flow is the path of an object as it rolls on the surface, starting from a high place and rolling down to a lower place (in the exact opposite direction as the gradient vectors point).
  • def from Streamline Tracing on Irregular Grids by H˚akon Hægland
    • "The instantaneous curves that are at every point tangent to the direction of the velocity at that point are called streamlines of the flow"
    • "A pathline of a fluid particle is the locus of its position in space as time passes. It is thus the trajectory of a particle of fixed identity"

Khan

the gradient points in the direction which increases the value of f most quickly. There are two ways to think about this direction:

  • Choose a fixed step size, and find the direction such that a step of that size increases fff the most. Given steps of a constant size away from a particular point, the gradient is the one which increases f the most.
  • Choose a fixed increase in fff, and find the direction such that it takes the shortest step to increase fff by that amount. Given steps which increase f by a given size, the gradient direction is the shortest among these.

Either way, you're trying to :

  • maximize the rise over run,
  • either by maximizing the rise, or minimizing the run.

MathWorks

Numerical Gradient The numerical gradient of a function is a way to estimate the values of the partial derivatives in each dimension using the known values of the function at certain points.

For a function of two variables, F(x,y), the gradient is

∇F=∂F/ ∂x ˆi + ∂F ∂y ˆ j  .

The gradient can be thought of as a collection of vectors pointing in the direction of increasing values of F. In MATLAB®, you can compute numerical gradients for functions with any number of variables.

Tips Use diff or a custom algorithm to compute multiple numerical derivatives, rather than calling gradient multiple times.

Algorithms gradient calculates the central difference for interior data points. For example, consider a matrix with unit-spaced data, A, that has horizontal gradient G = gradient(A). The interior gradient values, G(:,j), are

G(:,j) = 0.5*(A(:,j+1) - A(:,j-1)); The subscript j varies between 2 and N-1, with N = size(A,2).

gradient calculates values along the edges of the matrix with single-sided differences:

G(:,1) = A(:,2) - A(:,1); G(:,N) = A(:,N) - A(:,N-1); If you specify the point spacing, then gradient scales the differences appropriately. If you specify two or more outputs, then the function also calculates differences along other dimensions in a similar manner. Unlike the diff function, gradient returns an array with the same number of elements as the input.

potential flow

Construction of a potential flow.svg

spiral

The Golden Ratio and the Golden Angle

In disc phyllotaxis, as in the sunflower and daisy, the mesh of spirals occurs in Fibonacci numbers because divergence (angle of succession in a single spiral arrangement) approaches the golden ratio. The shape of the spirals depends on the growth of the elements generated sequentially. In mature-disc phyllotaxis, when all the elements are the same size, the shape of the spirals is that of Fermat spirals—ideally. That is because Fermat's spiral traverses equal annuli in equal turns. The full model proposed by H Vogel in 1979[5] is

where θ is the angle, r is the radius or distance from the center, and n is the index number of the floret and c is a constant scaling factor. The angle 137.508° is the golden angle which is approximated by ratios of Fibonacci numbers.[6]


Illustration of Vogel's formula of the pattern of sunflower florets (see article) for n from 1 to 500, using the polar coordinates equations and . Can be produced using the following MATLAB code:

n=1:500;
r=sqrt(n);
t=2*pi/((sqrt(5)+1)/2+1)*n;
plot(r.*cos(t),-r.*sin(t),'o')


https://www.shadertoy.com/view/4lGfDd https://www.youtube.com/watch?v=sj8Sg8qnjOg



Vogel's mode

A model for the pattern of florets in the head of a sunflower was proposed by Helmut Vogel in 1979.[7] This has the form

 
 
 

where:

  • n is the index number of the floret
  • c is a constant scaling factor

The florets thus lie on Fermat's spiral. The divergence angle, approximately 137.51°, is the golden angle, dividing the circle in the golden ratio. Because this ratio is irrational, no floret has a neighbor at exactly the same angle from the center, so the florets pack efficiently. Because the rational approximations to the golden ratio are of the form F(j):F(j + 1), the nearest neighbors of floret number n are those at n ± F(j) for some index j, which depends on r, the distance from the center. Sunflowers and similar flowers most commonly have spirals of florets in clockwise and counter-clockwise directions in the amount of adjacent Fibonacci numbers,[8] typically counted by the outermost range of radii.[9]


Algorithm

code

R code

#http://www.mathrecreation.com/2015/08/simple-fun-with-r.html
# https://github.com/dmackinnon1/r_examples/blob/master/simple/example2.r

#example 2b phyllotaxis spiral
t <- 1:500
p <- (1 + sqrt(5))*pi
plot(sqrt(t)*cos(p*t), sqrt(t)*sin(p*t), type="p", axes=FALSE)

Maple code

# code from : http://personal.maths.surrey.ac.uk/ext/R.Knott/Fibonacci/seedPlotMaple.txt
# by Ron Knott

> with(plots):

#Growpts shows a single picture (plot) of n seeds (points) distributed at 
#TurnperSeed (a numberbetween 0 and 1) which is the fraction of 1 turn between 
#one seed and the next,

> 
growpts:=(n,TpS)->growpts1(n,TpS,POINT):
growpts1:=proc(n,TurnperSeed,symb) local i,a,r,s,phi2pi;
   s:=null;
   phi2pi:=TurnperSeed*2*Pi;
   listplot([seq([sqrt(n-i)*cos(phi2pi*i),sqrt(n-i)*sin(phi2pi*i)],i=1..n)],
      style=POINT,axes=NONE,scaling=CONSTRAINED,symbol=symb)
end;



#Here is a seed-head with Pi turns between each seed.  
#Since Pi=3.14159>1, it is the same as 0.14159 turns per seed.  
#Note how there are 7 radial arms (corresponding to 22/7 for Pi) near the centre 
#and the next set of radial arms are 113 arms with seeds placed 16 arms apart 
#(since the next best approximation to Pi is 3+16/113=355/113).

> growpts(1000,Pi);

#Here we take a single Turns-per-seed value and keep adding a new seed 
#(at the centre) showing the seed head growing up to n seeds finally.  
#The plots are animated to show the growing process:


>
seedplot := proc(n, ratio)
  display([seq(growpts1(i, ratio, CIRCLE), i = (1 .. n))], insequence = true, 
  style = point, scaling = constrained, axes = NONE)
end proc;

#Here is 100 seeds at Phi =1.618.. turns per seed (which is the same as 
#Phi-1=0.618..=phi turns per seed):

> seedplot(100,(sqrt(5)-1)/2);

intersection of polar curves

iteration

atan2

Atan2 circle.svg



conformal

numerical conformal mapping=

Orthogonal

Ancillary figure for van Roomen's solution to the problem of Apollonius.

tangent to circle in the polar form

 z= r(t)

is line :

 y = x*tan(t)

Tangent to cardioid in polar form :

 z = 2a( 1 - cos(t))

is line :

 // y = (x*(-1+2*cos(t))*sin(t)+(-3+3*cos(t))*sin(t))/(-1+2*cos(t)^2-cos(t))
 y = (x*(-1+2*cos(t))*sin(t)+(-2a+2a*cos(t))*sin(t))/(-1-cos(t)+2*cos(t)^2)

where t is changing from 0 to 2*pi

Compare :


ellipse

slope m



The equation of the tangent at point has the form


https://www.math24.net/tangent-normal-lines-page-2/#example11 https://www.math24.net/implicit-differentiation/


Implicit

  • function
  • implicit differentiation = differentiation of the implicit function

sqare root

repeated

smooth curve from points

"curve fitting is a set of techniques used to fit a curve to data points "

Fit method

  • linear
    • join points with segments = concatenated linear segments
    • straight line using linear regression
  • nonlinear
    • polynomial
    • cubic spline
      • Smooth Bézier Spline Through Prescribed Points

trace a curve

sketch a curve

boundary trace

level sets

tree

People

Marius-F. Danca

petal

"An attracting petal, P + , for a map M at zero is an open simply connected forward invariant region with 0 ∈ ∂P + , that shrinks down to the origin under iteration of M . More precisely, P + is an attracting petal if M (P + ) ⊂ P + ∪ {0} and n≥0 M n (P + ) = {0}. "[10]


example

http://mathoverflow.net/questions/104482/parabolic-immediate-basins-always-simply-connected?rq=1 "An example is f(z)=z+1−1/zf(z)=z+1−1/z. There is one petal for the neutral point at infinity. Let AA be the dmain of attraction of ∞∞. Critical points are ±i±i. Everything is symmetric with respect to the real line, because the function is real. One critical point is in AA, so by symmetry the other one is also in AA. The map f:A→Af:A→A is 2-to-1 (because ff is of degree 22), so Riemann and Hurwitz tell us that AA is infinitely connected."

shareciteeditflag answered Aug 12 '12 at 13:38

Alexandre Eremenko

cylinder

What is the difference between the cylinder and cylinder  ?

topology

"A topologist is someone who doesn't know the difference between a cup of coffee and a donut."



El método de Mandelbrot : este método para desarrollar "objetos fractales" fue creado por Benoît Mandelbrot en la década de los años 70, mientras trabajaba en IBM. Consiste en construir, para cada punto c del plano complejo, una sucesión de números complejos zn. Partiendo del punto z0 = 0, se calcula la sucesión de forma iterativa mediante la fórmula zn+1=F(zn)+c, donde F es una función arbitraria previamente elegida. Cuando la sucesión iterativa está acotada, se asigna al punto c del plano complejo un color sólido (por ejemplo, el color negro). Si la sucesión diverge entonces se asigna al punto c un color progresivamente distinto, dependiendo de cuántas iteraciones hayan sido necesarias para detectar la divergencia de la sucesión.

El fractal derivado por este método cuando se toma la función F(z)=z2 se llama conjunto de Mandelbrot.

En lo que sigue, en lugar de zn+1=F(zn)+c se utilizará la notación Z=F(Z)+C, como si se tratara de una asignación en algún lenguaje de programación.

Z = Zm + C

A continuación se muestra una serie de fractales iterando las diferentes potencias de Z = Zm + C, según el método de Mandelbrot.

Ejemplos de fractales del tipo Mandelbrot: Z = Zm + C

Tal y como se puede ver en los ejemplos representados, el número de lóbulos es L = m - 1

Un breve viaje a las profundidades del fractal de Mandelbrot Z = Z2 + C

A continuación vamos a adentrarnos en el fractal clásico de Mandelbrot, utilizando el microscopio de altísima resolución que nos proporciona el cálculo iterativo. Todas las ampliaciones vienen precedidas de una imagen del fractal a escala 1:1 en donde podemos apreciar la zona ampliada.[11]

Ampliación zona 1

Centro de coordenadas : Cx = 0.291811 , Cy = 0.0144686

Ampliación zona 2

Centro de coordenadas : Cx = -0.165643411 , Cy = 0.656685704

Ampliación zona 3

Centro de coordenadas : Cx = -0.755625 , Cy = 0.06328125

Ampliación zona 4

Centro de coordenadas : Cx = -0,1758752481899, Cy = 1,075392007
A continuación bajaremos a gran profundidad, con una ampliación de más de 2 millones y con un número máximo de 6000 iteraciones por pixel !

Ampliación zona XX

Centro de coordenadas : Cx = 0,02816835288421, Cy = 0,63790834667330
Ahora nos adentraremos en un sitio con extrañas formas y colores, pero donde pueden apreciarse perfectamente las formas del fractal de Mandelbrot...

Z = Z-m + C

Ejemplos de fractales del tipo Mandelbrot, con potencias negativas de Z.

Z = Zp / (1 + Zq) + C

Z = Zm + Cp

Pero, ¿ qué pasa cuando hacemos Z = Zm + Cp ?. Tal y como se puede ver en los siguientes ejemplos, el número de lóbulos es L = (m - 1) * p

Z = Zm + Z + C

A continuación se muestra una serie de fractales iterando las diferentes potencias de Z = Zm + Z + C, según el método de Mandelbrot.

Z = Zm - Z + C

A continuación se muestra una serie de fractales iterando las diferentes potencias de Z = Zm - Z + C, según el método de Mandelbrot.

Z = Zm + 1 / Cp

También se puede transformar cada punto del plano complejo, de acuerdo a una función arbitraria, antes de ser sumado a la función iterativa, según la siguiente ecuación Z = Zm + F(C) . Veamos que pasa cuando la transformación es del tipo:F(C) = 1 / C

Ejemplos de fractales del tipo Mandelbrot: Z = Zm + 1/C, donde cada punto C del plano complejo se transforma en 1 / C, antes de entrar en la iteración de la potencia de Z.
Zo = (0,0i). El número de vértices es V = (m - 1)


Pero, qué pasa cuándo Z = Zm + (1 / C2) ?. Pues algo muy parecido a lo que veíamos antes, ahora el número de vértices es V = (m - 1) * p




Integrando en el mismo fractal una función de C y su inversa Z = Zm + C i Z = Zm + 1/C

La zona en color BLANCO intenso es el área de la intersección de los 2 sets.

Z = ( Zm / Cm ) + C

Z = Zm + C + Cp + 1/ C + 1/ Cq

También podemos añadir más sumandos a la función Zm, combinando C, Cp, 1/C y 1/Cq en grupos de 2, 3 o 4, veamos que sucede si agrupamos C2, 1/C y 1/C2 de 2,3 o 4 formas ..:


A continuación más combinaciones con otros exponentes:

Z = Zm + polinomios de C

Podemos combinar diferentes potencias de C y/o Z sumándolas a Zm , veamos qué sucede:

El caso de la función: Z=Z2 + 1 /(Cm-1)

Z = Zm + polinomios mixtos de C i Z

Podemos sumar a Zm polinomios mixtos de C i Z , veamos qué sucede:

Z = Z2 + C/ (Z2 + k)

Z = Zm + Cp/Zq + C

Z= [(Zm+C-1) / (m*Zm-1+C- m)]2

Z= [(Z + Cm-1) / Cm]2

Z= [(Z + Cm-1) / Cm]3

Z= [(Z + Cm+1) / (Cm - 1)]2

Z= [(Z + Cm-1) / (Cm + 1)]2

Otras combinaciones de Z y C

Más funciones de variable compleja

Pero existe una amplia variedad de funciones, en el dominio de los números complejos, que pueden ser iteradas según el método de Mandelbrot.
Voy a citar aquí algunos ejemplos, explicitando la parte real y la imaginaria:

Exp(Z) = [ Exp(x) * Cos(x), Exp(x) * Sin(y)i ]
Sin(Z) = [ Sin(x) * ((Exp(y) + Exp(-y)) / 2), Cos(x) * ((Exp(y) - Exp(-y)) / 2)i ]
Cos(Z) = [ Cos(x) * ((Exp(y) + Exp(-y)) / 2) , -Sin(x) * ((Exp(y) - Exp(-y)) / 2)i ]
SinH(Z) = [ Cos(y) * ((Exp(x) - Exp(-x)) / 2) , Sin(Y) * ((Exp(x) + Exp(-x)) / 2)i ]
CosH(Z) = [ Cos(y) * ((Exp(x) + Exp(-x)) / 2) , Sin(y) * ((Exp(x) - Exp(-x)) / 2)i ]
LN(Z) = [ 0.5 * Log(x * x + y * y) , Atn(y / x)i ]
SQR(Z) = [ (x * x + y * y)^0.25 * Cos(0.5 * Atn(y/x)) , (x * x + y * y)^0.25 * Sin(0.5 * Atn(y/x)) i ]
ATN(Z) = [PI / 4 - (1 / 2) * Atn((1 - x^2 - y^2) / (2 * x)), -(1 / 4) * Log((1 - x^2 - y^2) ^2 + 4 * x^2) + (1 / 2) * Log((1 + y) ^2 + x^2) i]


Z = Zm + F(C)

A continuación algunos ejemplos de fractales por iteración de Z2, pero transformando C según las funciones descritas anteriormente:

Fractales por iteración de Exp(Z)

Esta función se descompone en una parte real y otra imaginaria: Exp(Z) = [ Exp(x) * Cos(y), Exp(x) * Sin(y)i ]
Puede ser utilizada como función iterativa o como función transformadora de C = (Cx,Cyi), o simultáneamente:

Como función iterativa

Como función transformadora de C

Como función iterativa i transformadora de C, simultáneamente

El caso de la función Z = Exp[(Z2 + k * Z) / F(Cm)]

Esta función es muy sensible a Zo, y también al coeficiente (k) que multiplica a Z. Veamos algunos ejemplos interesantes:

El caso de la función Zn+1 = Exp(Zn / C m)


El caso de la función Zn+1 = Exp(Zn / C m) + C p

El caso de la función Zn+1 = Exp(Znp / C p)


El caso de la función Zn+1 = Znq * Exp(Zn / C p) + C

El caso de la función Zn+1 = Exp[ Zn2 / (C m + C p) ]

Aparece un número de lóbulos centrales = m, y un número de aristas exteriores = p, siendo m<p.


El caso de la función Zn+1 = Znm * Exp[ Cos(Zn)] + 1/C

Aparecen un número de aristas = m.

Fractales per iteración de Sin(Z)

Esta función se descompone en una parte real y otra imaginaria: Sin(Z) = [ Sin(x) * ((Exp(y) + Exp(-y)) / 2), Cos(x) * ((Exp(y) - Exp(-y)) / 2)i ]
Puede ser utilizada como función iterativa o como función transformadora de los puntos C = (Cx,Cyi), simultáneamente:

Como función iterativa

Como función transformadora de C


Como función iterativa y transformadora de C, simultáneamente

El caso de la función Zn+1 = Sin(Zn * C m)

El caso de la función Zn+1 = Sin(Zn / C m)

Fractales por iteración de Cos(Z)

Esta función se descompone en una parte real y otra imaginaria: ' Cos(Z) = [ Cos(x)*((Exp(y)+Exp(-y)) / 2), -Sin(x)*((Exp(y)-Exp(-y))/2)i ]
Puede ser utilizada como función iterativa o como función transformadora de C = (Cx,Cyi), o simultáneamente

Como función iterativa

Como función transformadora de C

Como función iterativa y transformadora de C, simultáneamente

El caso de la función Zn+1 = Cos(Zn * C m)


El caso de la función Zn+1 = Cos(Zn/C m)


Fractales por iteración de SinH(Z)

Esta función se descompone en una parte real y otra imaginaria: SinH(Z) = [ Cos(y) * ((Exp(x) - Exp(-x)) / 2) , Sin(Y) * ((Exp(x) + Exp(-x)) / 2)i ]
Puede ser utilizada como función iterativa o como función transformadora de C = (Cx,Cyi), o simultáneamente:

Como función iterativa

Como función transformadora de C


Como función iterativa i transformadora de C, simultáneamente



Fractales por iteración de CosH(Z)

Esta función se descompone en una parte real y otra imaginaria: CosH(Z) = [ Cos(y) * ((Exp(x) + Exp(-x)) / 2) , Sin(y) * ((Exp(x) - Exp(-x)) / 2)i ]
Puede ser utilizada como función iterativa o como función transformadora de C = (Cx,Cyi), o simultáneamente:

Como función iterativa

Fractales por iteración de combinaciones de diferentes funciones de Z

Más fractales según el método de Mandelbrot

Aquí se muestra un ejemplo de iteración de dos funciones F(X) y F(Y), por adición de cada uno de los puntos del plano C(X,Y), y la introducción de una tercera función F(Z) que desequilibra el punto de convergencia.
Xn+1 = Xn - Sin(Yn) + C(X) .. Yn+1 = Yn - Sin( Xn) + C(Y) .. Zn+1 = Zn - Cos( Xn + Yn)

references

  • Computer Methods and Borel Summability Applied to Feigenbaum's Equation By Jean Pierre Eckmann

Format Hardback | 297 pages, Publication date 01 May 1985, Publisher Springer , Publication City/Country United States , ISBN10 0387152156, ISBN13 9780387152158

  • http://mathoverflow.net/questions/157309/power-series-expansion-of-the-koenigs-function?rq=1
  • T. M. CHERRY, A singular case of iteration of analytic functions: A contribution to the small divisor problem. In: Nonlinear Problems of Engineering, W. F. AMES (Ed.), New York, 1964, 29–50.
  • Cherry, T. M., "A Singular Case of Iteration of Analytic Functions: A Contribution to the Small-Divisor Problem," in Nonlinear Problems of Engineering (edited by W. F. Ames), Academic Press, New York, 1964, 29-50.
  • MR178125 30.40 (57.48) Cherry, T. M. A singular case of iteration of analytic functions: A contribution to the small-divisor problem. 1964 Nonlinear Problems of Engineering pp. 29–50 Academic Press, New York
pl-N Polski jest językiem ojczystym tego użytkownika.
en-2 This user can read and write intermediate English.

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  3. fractalforums.org : line-segments-intersecting-msets-how-often
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  5. Vogel, H (1979). "A better way to construct the sunflower head". Mathematical Biosciences 44 (44): 179–189. doi:10.1016/0025-5564(79)90080-4 
  6. Prusinkiewicz, Przemyslaw; Lindenmayer, Aristid (1990). The Algorithmic Beauty of Plants. Springer-Verlag. pp. 101–107. ISBN 978-0-387-97297-8. http://algorithmicbotany.org/papers/#webdocs. 
  7. Vogel, Helmut (1979), "A better way to construct the sunflower head", Mathematical Biosciences 44 (3–4): 179–89, doi:10.1016/0025-5564(79)90080-4 
  8. Livio 2003, p. 112.
  9. Prusinkiewicz, Przemyslaw; Lindenmayer, Aristid (1990), "4", The Algorithmic Beauty of Plants, Springer-Verlag, pp. 101–107, ISBN 978-0-387-97297-8, http://algorithmicbotany.org/papers/#webdocs 
  10. NEWTON’S METHOD ON THE COMPLEX EXPONENTIAL FUNCTION MAKO E. HARUTA
  11. Barnsley, M. Fractals everywhere.Academic Press Inc, 1988. ISBN 0-12-079062-9. (Cap 5)