  "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:"

My home page - dead (:-(

Wikipedia - Adam majewski

Commons - Adam majewski

My Dropbox public folders

Mathematics Stack Exchange

gitlab repo

Links :

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

icc

  convert image_rgb.tiff -profile "RGB.icc" -profile "CMYK.icc" image_cmyk.tiff

graphic software

text color

some notes

Some notes from | wikipedia

$Z^{2}$

$\operatorname {Re} =x^{2}-y^{2}$
$\operatorname {Im} =2*x*y$

$Z^{3}$

$\operatorname {Re} =x^{3}-3*y^{2}*x$
$\operatorname {Im} =3*x^{2}*y-y^{3}$

$Z^{4}$

$\operatorname {Re} =x^{4}-6*x^{2}*y^{2}+y^{4}$
$\operatorname {Im} =4*x^{3}*y-4*x*y^{3}$

$Z^{5}$

$\operatorname {Re} =x^{5}-10*x^{3}*y^{2}+5*x*y^{4}$
$\operatorname {Im} =5*x^{4}*y-10*x^{2}*y^{3}+y^{5}$

$Z^{6}$

$\operatorname {Re} =x^{6}-15*x^{4}*y^{2}+15*x^{2}*y^{4}-y^{6}$
$\operatorname {Im} =6*x^{5}*y-20*x^{4}*y^{2}+6*x*y^{5}$

$Z^{7}$

$\operatorname {Re} =x^{7}-21*x^{5}*y^{2}+35*x^{3}*y^{4}-7*x*y^{6}$
$\operatorname {Im} =7*x^{6}*y-35*x^{4}*y^{3}+21*x^{2}*y^{5}-y^{7}$ ,

$\exp(Z)$

$\operatorname {Re} =\exp(x)*\cos(y)$
$\operatorname {Im} =\exp(x)*\sin(y)$

$\ln(Z)$

$\operatorname {Re} =0.5*\ln(x^{2}+y^{2})$
$\operatorname {Im} =\arctan(y/x)$

$\sin(Z)$

$\operatorname {Re} =\sin(x)*((\exp(y)+\exp(-y))/2)$
$\operatorname {Im} =\cos(x)*((\exp(y)-\exp(-y))/2)$

$\cos(Z)$

$\operatorname {Re} =\cos(x)*((\exp(y)+\exp(-y))/2)$
$\operatorname {Im} =-\sin(x)*((\exp(y)-\exp(-y))/2)$

$\sinh(Z)$

$\operatorname {Re} =\cos(y)*((\exp(x)-\exp(-x))/2)$
$\operatorname {Im} =\sin(y)*((\exp(x)+\exp(-x))/2)$

$CosH(Z)$

$\operatorname {Re} =\cos(y)*((\exp(x)+\exp(-x))/2)$
$\operatorname {Im} =\sin(y)*((\exp(x)-\exp(-x))/2)$

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^[3]

PDE ( Linear Second Order PDE’s in two Independent Variables) : https://en.wikipedia.org/wiki/Partial_differential_equation
Moebius transformations = Classification of Isometries ( https://www.mathi.uni-heidelberg.de/~alessandrini/Arith_Reports/1-hyperbolic%20geometry.pdf)
dicrete local complex dynamics
Conic section
Quadratic form
probability distributions.

coordinate
elliptical

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

https://www.geogebra.org/m/eemfktww

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)

computer graphic

www.pling.org.uk/cs/cgv.html
www.tutorialspoint.com/computer_graphics/computer_graphics_curves.htm
github.com/jagregory/abrash-black-book
  pages.mtu.edu/~shene/COURSES/cs3621/NOTES/model/b-rep.html
  pages.mtu.edu/~shene/COURSES/cs3621/NOTES/notes.html

libraries

graphics pipeline

https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-837-computer-graphics-fall-2012/lecture-notes/MIT6_837F12_Lec21.pdf

geometry

Moore-Neighbor Tracing

chain code

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).
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.

potential flow

https://gregjavens.com/2016/03/12/laplace-equation-finite-element-method-3/

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^[4] 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.^[5]

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')

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

intersection of polar curves

iteration

atan2

conformal

" Conformal transformation: One can find solutions to the Dirichlet problem for a wider class of regions by conformal transformations. These are transformations which preserve the angles and have the property that the transplanted function is harmonic if the original function is harmonic. One can obtain conformal transformations using complex function (x,y) -> (u,v): if z=x+iy and F(z) is a polynomial in z and F(z)=u+i v, then the map is conformal. For example, to solve the Dirichlet problem outside a wing, one can transform the circle into the wing using a conformal transformation. Such methods are relevant in engeneering." http://www.math.harvard.edu/archive/21b_fall_03/laplace/index.html
https://www3.cs.stonybrook.edu/~gu/tutorial/RiemannMapping.html
https://www.chebfun.org/docs/guide/guide16.html
https://www.cs.cmu.edu/~kmcrane/Projects/Other/OverviewConformalGeometryProcessing.pdf
http://www-users.math.umn.edu/~olver/ln_/cml.pdf
The Euler conformal map, a special case of the M¨obius transformation, maps circles to circles
https://gregjavens.com/2016/05/11/conformal-map/
https://gregjavens.com/2016/03/12/laplace-equation-finite-element-method-3/
http://mathfaculty.fullerton.edu/mathews/c2003/ConformalMapDictionary.2.html
https://www3.cs.stonybrook.edu/~gu/
https://arxiv.org/abs/1306.1162
http://www.eng.biu.ac.il/weberof/publications/
http://wwwf.imperial.ac.uk/~dgcrowdy/GuTalk.pdf

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)

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

Compare :

ellipse

$(x,y)=(a\cos t,b\sin t),\ 0\leq t<2\pi \ .$

slope m

$m=-{\frac {b}{a}}\cot t\quad$

The equation of the tangent at point ${\vec {c}}_{\pm }(m)$ has the form $y=mx+n$

$y=mx\pm {\sqrt {m^{2}a^{2}+b^{2}}}\;.$

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

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

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

http://xaktly.com/DerivativesIII.html

boundary trace

level sets

https://scicomp.stackexchange.com/questions/1348/given-values-on-a-mesh-what-algorithm-can-i-use-to-construct-efficiently-level?rq=1

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}. "^[6]

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
http://www.scholarpedia.org/article/Hyperbolic_dynamics

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.

.

↑ math SE question: is-the-basin-of-attraction-of-a-p-starshaped-wrt-to-p
↑ Why Category Theory Matters by Robb Seaton
↑ quora : Where-is-the-best-summary-on-the-meaning-of-the-terms-elliptic-hyperbolic-parabolic-as-used-in-different-disciplines-in-mathematics
↑ 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] th SE question: is-the-basin-of-attraction-of-a-p-starshaped-wrt-to-p

[2] Why Category Theory Matters by Robb Seaton

[3] quora : Where-is-the-best-summary-on-the-meaning-of-the-terms-elliptic-hyperbolic-parabolic-as-used-in-different-disciplines-in-mathematics

[4] 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

[5] 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.

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

[1]

[2]

[3]

[4]

[5]

[6]

User:Adam majewski

rigorous studies of nonlinear systems

methods

icc

graphic software

text color

some notes

parabolic/hyperbolic/elliptic

curves

curvature

topology

number theory

algorithms

computer graphic

libraries

graphics pipeline

geometry

Moore-Neighbor Tracing

chain code

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

conformal

numerical conformal mapping=

Orthogonal

ellipse

sqare root

repeated

smooth curve from points

trace a curve

sketch a curve

boundary trace

level sets

tree

People

petal

example

cylinder

topology

hyperbolic

references

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