  " 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.

Hi,

Wikipedia - Adam majewski

Commons - Adam majewski

My Dropbox public folders

Mathematics Stack Exchange

gitlab repo

Links :

Some notes from | wikipedia

$Z^{2}$ $Real=x^{2}-y^{2}$ $Imag=2*x*y$

$Z^{3}$ $Real=x^{3}-3*y^{2}*x$ $Imag=3*x^{2}*y-y^{3}$

$Z^{4}$ $Real=x^{4}-6*x^{2}*y^{2}+y^{4}$ $Imag=4*x^{3}*y-4*x*y^{3}$

$Z^{5}$ $Real=x^{5}-10*x^{3}*y^{2}+5*x*y^{4}$ $Imag=5*x^{4}*y-10*x^{2}*y^{3}+y^{5}$

$Z^{6}$ $Real=x^{6}-15*x^{4}*y^{2}+15*x^{2}*y^{4}-y^{6}$ $Imag=6*x^{5}*y-20*x^{4}*y^{2}+6*x*y^{5}$

$Z^{7}$ $Real=x^{7}-21*x^{5}*y^{2}+35*x^{3}*y^{4}-7*x*y^{6}$ $Imag=7*x^{6}*y-35*x^{4}*y^{3}+21*x^{2}*y^{5}-y^{7}$ ,

$Exp(Z)$ $Real=Exp(x)*Cos(y)$ $Imag=Exp(x)*Sin(y)$

$Ln(Z)$ $Real=0.5*Ln(x^{2}+y^{2})$ $Imag=Arctan(y/x)$

$Sin(Z)$ $Real=Sin(x)*((Exp(y)+Exp(-y))/2)$ $Imag=Cos(x)*((Exp(y)-Exp(-y))/2)$

$Cos(Z)$ $Real=Cos(x)*((Exp(y)+Exp(-y))/2)$ $Imag=-Sin(x)*((Exp(y)-Exp(-y))/2)$

$SinH(Z)$ $Real=Cos(y)*((Exp(x)-Exp(-x))/2)$ $Imag=Sin(y)*((Exp(x)+Exp(-x))/2)$

$CosH(Z)$ $Real=Cos(y)*((Exp(x)+Exp(-x))/2)$ $Imag=Sin(y)*((Exp(x)-Exp(-x))/2)$

test

$1/237142198758023568227473377297792835283496928595231875152809132048206089502588927\approx 4.21687917729220928973942962050800760308398455294740302003110521004325771638790385468222014406044810316697454753662...*10^{-}81$ ${\frac {1}{2^{267}-1}}={\frac {1}{237142198758023568227473377297792835283496928595231875152809132048206089502588927}}\approx 4.216879177292209*10^{-81}$ ( land on the root point of period 267 component : c267 = 0.250137369683480-0.000003221184145 i with angled internal adress : $1{\xrightarrow {1/267}}267$

( land on the root point of period 268 component c268 = 0.250137369683480-0.000003221184145i period = 10000 i with angled internal adress : $1{\xrightarrow {267/268}}268$

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":

  $f(z)={\frac {1}{z}}-{\frac {z*(1+0.001*z)}{(1-0.002*z+0.001*z^{2})}}$

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).

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.

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^[1] is

r=c{\sqrt {n}},

\theta =n\times 137.508^{\circ },

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.^[2]

The pattern of florets produced by Vogel's model (central image). The other two images show the patterns for slightly different values of the angle.

Illustration of Vogel's formula of the pattern of sunflower florets (see article) for n from 1 to 500, using the polar coordinates equations $r=c{\sqrt {n}}$ and $\theta =n\times {\frac {2\pi }{\phi +1}}$ . 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')

intersection of polar curves

iteration

atan2

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)

Creating a cardioid by rolling a circle on a circle of the same radius
animation

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

sqare root

http://mathlets.org/mathlets/complex-roots/
https://flothesof.github.io/branch-cuts-with-square-roots.html
http://phantomgraphs.weebly.com/
https://math.stackexchange.com/questions/1797223/graphically-solving-for-complex-roots-how-to-visualize?noredirect=1&lq=1
Reciprocal Function as a Mapping
branch cuts
- https://www.cs.cmu.edu/Groups/AI/html/cltl/clm/node129.html
- sqrt : The branch cut for square root lies along the negative real axis, continuous with quadrant II. The range consists of the right half-plane, including the non-negative imaginary axis and excluding the negative imaginary axis.
- https://math.stackexchange.com/questions/923931/problem-identifying-branch-cuts-of-a-square-root-function?noredirect=1&lq=1

smooth curve from 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

tree

https://www.youtube.com/watch?v=BEz-vGJvaik

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}. "^[3]

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 $\mathbb {C} \setminus 0$ and cylinder $\mathbb {C} /\mathbb {Z}$ ?

topology

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

hyperbolic

hyperbolic". usually) means that |f′(t)|≠1| , https://math.stackexchange.com/questions/2172002/is-indeterminate-a-better-name-than-indifferent-for-neutral-fixed-points

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.

.

↑ 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
↑ 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.
↑ NEWTON’S METHOD ON THE COMPLEX EXPONENTIAL FUNCTION MAKO E. HARUTA

[1] 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

[2] 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.

[3] NEWTON’S METHOD ON THE COMPLEX EXPONENTIAL FUNCTION MAKO E. HARUTA

[1]

[2]

[3]

User:Adam majewski

Contents

test

mandelbrot set

video

programs

gradient line of the 2D scalar field

Khan

potential flow

spiral

The Golden Ratio and the Golden Angle

intersection of polar curves

iteration

atan2

Orthogonal

sqare root

smooth curve from points

tree

People

petal

example

cylinder

topology

hyperbolic

references

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