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What (who) is quadratic complex - definition

QUADRATIC POLYNOMIAL

Quadratic map; Complex quadratic map

$* f_c planes (all other planes for c \ne 0)$
$: z_n' = \frac{d}{dc} f_c^n(z_0).$
$: z_0' = \frac{d}{dc} f_c^0(z_0) = 1$
$: z_{n+1}' = \frac{d}{dc} f_c^{n+1}(z_0) = 2\cdot{}f_c^n(z)\cdot\frac{d}{dc} f_c^n(z_0) + 1 = 2 \cdot z_n \cdot z_n' +1.$
$: f_c'(z_0) = \frac{d}{dz}f_c(z_0) = 2z_0 .$
$: (f_c^p)'(z_0) = \frac{d}{dz}f_c^p(z_0) = \prod_{i=0}^{p-1} f_c'(z_i) = 2^p \prod_{i=0}^{p-1} z_i = \lambda$
$:z'_n= 2*z_{n-1}*z'_{n-1}.$
$: (Sf)(z) = \frac{f'''(z)}{f'(z)} - \frac{3}{2} \left ( \frac{f''(z)}{f'(z)}\right ) ^2 .$
$*[https://web.archive.org/web/20040828174339/http://www.math.sunysb.edu/cgi-bin/thesis.pl?thesis02-3 Wolf Jung : Homeomorphisms on Edges of the Mandelbrot Set. Ph.D. thesis of 2002]$

Parameter plane types> MandelbrotLambda.jpg|r parameter plane ( logistic map) MandelbrotMuDouadyRabbit.jpg| c parameter plane </gallery> The [[phase space]] of a quadratic map is called its '''parameter plane'''. Here: <math>z_0 = z_{cr}</math> is constant and <math>c</math> is variable. There is no dynamics here. It is only a set of parameter values. There are no orbits on the parameter plane. The parameter plane consists of: * The [[Mandelbrot set]] ** The [[bifurcation locus]] = boundary of [[Mandelbrot set]] with *** root points ** Bounded hyperbolic components of the Mandelbrot set = interior of Mandelbrot set<ref>[http://arxiv.org/abs/0805.1658 [[Lasse Rempe]], Dierk Schleicher : Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity]{{Dead link|date=July 2020 |bot=InternetArchiveBot |fix-attempted=yes }}</ref> with internal rays * exterior of Mandelbrot set with ** [[external ray]]s ** equipotential lines There are many different subtypes of the parameter plane.<ref>[http://aleph0.clarku.edu/~djoyce/julia/altplane.html Alternate Parameter Planes by David E. Joyce ]</ref><ref>[http://mrob.com/pub/muency/exponentialmap.html exponentialmap by Robert Munafo]</ref> [[File:Mandelbrot set - multiplier map.png|thumb|right|Multiplier map]] See also : * [[External ray#Uniformization 2|Boettcher map]] which maps exterior of Mandelbrot set to the exterior of unit disc * multiplier map which maps interior of hyperbolic component of Mandelbrot set to the interior of unit disc ====2D Dynamical plane==== <blockquote>

Complex quadratic polynomial

A complex quadratic polynomial is a quadratic polynomial whose coefficients and variable are complex numbers.

https://en.wikipedia.org/wiki/Complex_quadratic_polynomial

Quadratic irrational number

MATHEMATICAL CONCEPT

Quadratic surd; Quadratic irrationality; Quadratic Irrational Number; Quadratic irrationalities; Quadratic irrational; Quadratic irrational numbers

In mathematics, a quadratic irrational number (also known as a quadratic irrational, a quadratic irrationality or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational numbers.Jörn Steuding, Diophantine Analysis, (2005), Chapman & Hall, p.

https://en.wikipedia.org/wiki/Quadratic_irrational_number

Quadratic reciprocity

THEOREM

Law of quadratic reciprocity; Quadratic reciprocity rule; Aureum Theorema; Law of Quadratic Reciprocity; Quadratic reciprocity law; Quadratic reciprocity theorem; Quadratic Reciprocity; Qr theorem

In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is:

https://en.wikipedia.org/wiki/Quadratic_reciprocity

Wikipedia

Complex quadratic polynomial

A complex quadratic polynomial is a quadratic polynomial whose coefficients and variable are complex numbers.

https://en.wikipedia.org/wiki/Complex quadratic polynomial