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Result of repeatedly applying a mathematical function
In mathematics, an iterated function is a function that is obtained by composing another function with itself two or several times. The process of repeatedly
Iterated_function
Method for the construction of fractals
In mathematics, iterated function systems (IFSs) are a method of constructing fractals; the resulting fractals are often self-similar. IFS fractals are
Iterated_function_system
Fractal sets in complex dynamics of mathematics
can cause drastic changes in the sequence of iterated function values. Thus the behavior of the function on the Fatou set is "regular", while on the Julia
Julia_set
Arithmetic operation
iterated exponentials, as it is common to call expressions of this form iterated exponentiation, which is ambiguous, as this can either mean iterated
Tetration
Inverse function to a tower of powers
universe), the iterated logarithm with base 2 has a value no more than 5. Higher bases give smaller iterated logarithms. The iterated logarithm is closely
Iterated_logarithm
Repetition of a process
of statements is said to be iterated. A computer programmer might also refer to that block of statements as an iteration. Loops constitute the most common
Iteration
Compression method for digital images
Fractal image representation may be described mathematically as an iterated function system (IFS). We begin with the representation of a binary image,
Fractal_compression
Root-finding algorithm
sequence x 0 , x 1 , x 2 , … {\displaystyle x_{0},x_{1},x_{2},\dots } of iterated function applications x 0 , f ( x 0 ) , f ( f ( x 0 ) ) , … {\displaystyle
Fixed-point_iteration
Fractal constructible with L-systems
left: The Heighway dragon is also the limit set of the following iterated function system in the complex plane: f 1 ( z ) = ( 1 + i ) z 2 {\displaystyle
Dragon_curve
Operation on mathematical functions
square root Functional equation Higher-order function Infinite compositions of analytic functions Iterated function Lambda calculus The strict sense is used
Function_composition
Property of operations
(mathematics) Iterated function List of matrices Nilpotent Pure function Referential transparency This is an equation between functions. Two functions are equal
Idempotence
Object that enables processing collection items in order
can also be directly iterated over, when the dictionary keys are returned; or the items() method of a dictionary can be iterated over where it yields
Iterator
Concept in calculus of variations
({\boldsymbol {r}}-{\boldsymbol {r}}').} The functional derivative of the iterated function f ( f ( x ) ) {\displaystyle f(f(x))} is given by: δ f ( f ( x ) )
Functional_derivative
Function that, applied twice, gives another function
discuss], or rather f = g 1/2 (see Iterated function), although this leaves the usual ambiguity with taking the function to that power in the multiplicative
Functional_square_root
Characterises an iterated function system whose attractor is close to a given set
In mathematics, the collage theorem characterises an iterated function system whose attractor is close, relative to the Hausdorff metric, to a given set
Collage_theorem
Function reducing distance between all points
the iterated function sequence x, f (x), f (f (x)), f (f (f (x))), ... converges to the fixed point. This concept is very useful for iterated function systems
Contraction_mapping
Process of repeating items in a self-similar way
angled mirrors reflecting each other Iterated function – Result of repeatedly applying a mathematical function Mathematical induction – Form of mathematical
Recursion
Fractal composed of triangles
This method is also called the chaos game, and is an example of an iterated function system. You can start from any point outside or inside the triangle
Sierpiński_triangle
Infinitely detailed mathematical structure
small change in a single variable can have an unpredictable outcome. Iterated function systems (IFS) – use fixed geometric replacement rules; may be stochastic
Fractal
On finding a repeating loop in a sequence
of iterated function values. For any function f that maps a finite set S to itself, and any initial value x0 in S, the sequence of iterated function values
Cycle_detection
collective action of a set of contractions, called an iterated function system. The iteration of the operator converges to a unique attractor, which
Hutchinson_operator
Functional square root of an exponential
for every C > 0 {\displaystyle C>0} . Iterated function – Result of repeatedly applying a mathematical function Schröder's equation – Equation for fixed
Half-exponential_function
Fractal creation method
attractor, or the fixed point, of any iterated function system (IFS). Starting with any point x0, successive iterations are formed as xk+1 = fr(xk), where
Chaos_game
Point which a function/system returns to after some time or iterations
of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations
Periodic_point
Piecewise function that clamps its input to be non-negative
\}}(s)=\int _{0}^{\infty }e^{-sx}R(x)dx={\frac {1}{s^{2}}}.} Every iterated function of the ramp mapping is itself, as R ( R ( x ) ) = R ( x ) . {\displaystyle
Ramp_function
Fractal functions in mathematics
Fractal flames are a member of the iterated function system class of fractals created by Scott Draves in 1992. Draves' open-source code was later ported
Fractal_flame
Visual representation of an iterated function
mathematics to investigate the qualitative behaviour of one-dimensional iterated functions, such as the logistic map. The technique was introduced in 1822 by
Cobweb_plot
Continuous fractal curve obtained as the image of Cantor space
an Iterated function system using the set of contraction mappings { d 0 , d 1 } {\displaystyle \{d_{0},\ d_{1}\}} . But the result of an iterated function
De_Rham_curve
Element mapped to itself by a mathematical function
sequence x 0 , x 1 , x 2 , … {\displaystyle x_{0},x_{1},x_{2},\dots } of iterated function applications x 0 , f ( x 0 ) , f ( f ( x 0 ) ) , … {\displaystyle
Fixed_point_(mathematics)
Functions of an angle
case, the superscript could be considered as denoting a composed or iterated function, but negative superscripts other than − 1 {\displaystyle {-1}} are
Trigonometric_functions
Curve whose range contains the unit square
endpoints) is a continuous function whose domain is the unit interval [0, 1]. In the most general form, the range of such a function may lie in an arbitrary
Space-filling_curve
Standard example in game theory
multi-player iterated version of the game. In 1975, Grofman and Pool estimated the count of scholarly articles devoted to it at over 2,000. The iterated prisoner's
Prisoner's_dilemma
Three-dimensional fractal
remaining smaller cubes and continue to iterate ad infinitum. The second iteration gives a level 2 sponge, the third iteration gives a level 3 sponge, and so on
Menger_sponge
Quickly growing function
iterated function. Meyer & Ritchie (1967) showed this correspondence. These considerations concern the recursion depth only. Either way of iterating leads
Ackermann_function
State of a dynamic system after an infinitely long time
) } n ∈ N {\displaystyle \{f^{n}(x)\}_{n\in \mathbb {N} }} of the iterated function f {\displaystyle f} . Hence, y ∈ ω ( x , f ) {\displaystyle y\in \omega
Limit_set
Mathematical theory about infinitely iterated function composition
venue for iteration of systems of functions rather than a single function. For infinite compositions of a single function see Iterated function. For compositions
Infinite compositions of analytic functions
Infinite_compositions_of_analytic_functions
Plane fractal built from squares
zero (in standard Lebesgue measure). Proof: Denote as ai the area of iteration i. Then ai + 1 = 8/9ai. So ai = (8/9)i, which tends to 0 as i goes
Sierpiński_carpet
Software generating fractal images
technique first proposed in 1904 by Koch. The other main method is with Iterated Function Systems consisting of a number of affine transformations. In the first
Fractal-generating_software
Theorem in order and lattice theory
noncontractive discontinuous (multivalued) iterated function systems. For weakly contractive iterated function systems the Kantorovich theorem (known also
Knaster–Tarski_theorem
Type of integral of functions of multiple variables
In multivariable calculus, an iterated integral is the result of applying integrals to a function of more than one variable (for example f ( x , y ) {\displaystyle
Iterated_integral
Use of functions that call themselves
{ traverse(); } } This code is both recursion and iteration - the files and directories are iterated, and each directory is opened recursively. The "rtraverse"
Recursion_(computer_science)
Function computable with bounded loops
iterations of every loop is fixed before entering the loop). Primitive recursive functions form a strict subset of those general recursive functions that
Primitive_recursive_function
October 2018. Duda, Jarek (March 2011). "The Boundary of Periodic Iterated Function Systems", Wolfram.com. Chang, Angel and Zhang, Tianrong. "On the Fractal
List of fractals by Hausdorff dimension
List_of_fractals_by_Hausdorff_dimension
Proposition in probability theory
in probability theory known as the law of total expectation, the law of iterated expectations (LIE), Adam's law, the tower rule, and the smoothing property
Law_of_total_expectation
Computer graphics images defined by points, lines and curves
parametric surfaces (e.g., NURBS). Fractals, often defined as an iterated function system. In many vector datasets, each shape can be combined with a
Vector_graphics
Software library for the C++ programming language
container such as a map or set can be much slower using iterators than by calling member functions offered by the container itself. This is because an associative
Standard_Template_Library
Volunteer distributed computing project
project relies on the "fractal flame" algorithm, an extension of iterated function systems (IFS) which Draves created and released as open-source software
Electric_Sheep
Condition for fractals in math
on the overlap in a fractal construction. Specifically, given an iterated function system of contractive mappings ψ 1 , … , ψ m {\displaystyle \psi _{1}
Open_set_condition
Arithmetic operation
composition, commonly called the nth iterate of the function. Thus f n {\displaystyle f^{n}} denotes generally the nth iterate of f; for example, f 3 ( x ) {\displaystyle
Exponentiation
Rewriting system and type of formal grammar
Wikimedia Commons has media related to L-systems. Digital morphogenesis Iterated function system Reaction–diffusion system – Type of mathematical model that
L-system
Concept in topology
important in the study of iterated functions and more generally dynamical systems, since, if the dynamics of one iterative function can be determined, then
Topological_conjugacy
Operator encoding information about iterated map
theorem to the determination of the eigenvalues of the operator. The iterated function to be studied is a map f : X → X {\displaystyle f\colon X\rightarrow
Transfer_operator
Repeated application of an operation to a sequence
In mathematics, an iterated binary operation is an extension of a binary operation on a set S to a function on finite sequences of elements of S through
Iterated_binary_operation
Doubling map on the unit interval
dyadic transformation can also be defined as the iterated function map of the piecewise linear function T ( x ) = { 2 x 0 ≤ x < 1 2 2 x − 1 1 2 ≤ x < 1
Dyadic_transformation
"Pushed forward" from one measurable space to another
{f\circ f\circ \dots \circ f} _{n\mathrm {\,times} }:X\to X.} This iterated function forms a dynamical system. It is often of interest in the study of
Pushforward_measure
Equation for function that computes iterated values
equation Böttcher's equation Infinite compositions of analytic functions Iterated function Shift operator Superfunction Aczél, János, (1966): Lectures on
Abel_equation
Fractals is a scaling algorithm based on the use of PIFS (partitioned iterated function systems). When scaling up, Genuine Fractals exploits the self-similarity
Genuine_Fractals
Description of limiting behavior of a function
use asymptotic analysis for computing function approximations, implicit functions, integrals, iterated functions, series summation, partial sums, solutions
Asymptotic_analysis
Limit type in multivariable calculus
In multivariable calculus, an iterated limit is a limit of a sequence or a limit of a function in the form lim m → ∞ lim n → ∞ a n , m = lim m → ∞ ( lim
Iterated_limit
is a nonstandard name for an iterated function for complexified continuous iteration index. Roughly, for some function f and for some variable x, the
Superfunction
Geometric figure
applications. These developments are suitable to perform aesthetic and practical functions that are defined in advance by the consciously selected arrangements of
Spidron
Whole of an object being mathematically similar to part of itself
structure. The homeomorphisms may be iterated, resulting in an iterated function system. The composition of functions creates the algebraic structure of
Self-similarity
Family of higher-order functions
fold, proving that iterations can be reduced to folds: y f = foldr (\_ -> f) undefined (repeat undefined) Aggregate function Iterated binary operation Catamorphism
Fold_(higher-order_function)
Topics referred to by the same term
particularly software development. It can also refer to: Iterated function, in mathematics "Iteration", a song from Potemkin City Limits, the fourth full-length
Iteration_(disambiguation)
Conditions for switching order of integration in calculus
if a function is Lebesgue integrable on a rectangle X × Y {\displaystyle X\times Y} , then one can evaluate the double integral as an iterated integral:
Fubini's_theorem
Topics referred to by the same term
Service, a short-lived animation studio owned by Hearst Communication. Iterated function system, a method of constructing fractals in mathematics and computer
IFS
Algorithm for finding zeros of functions
the function's root than the previous guess, and the method can be iterated. The best linear approximation to an arbitrary differentiable function f (
Newton's_method
Fractal which resembles a plant
mathematical models. The fern code developed by Barnsley is an example of an iterated function system (IFS) to create a fractal. This follows from the collage theorem
Barnsley_fern
Field of mathematics and science based on non-linear systems and initial conditions
ISBN 978-0-521-66385-4. Collet, Pierre; Eckmann, Jean-Pierre (1980). Iterated Maps on the Interval as Dynamical Systems. Birkhauser. ISBN 978-0-8176-4926-5
Chaos_theory
Continuous function that is not absolutely continuous
In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in
Cantor_function
Open subset of the real–number line
function carries geometric information about the underlying fractal, particularly in the location of its poles and the residues of the zeta function at
Fractal_string
computer-assisted composition, and in particular iterated function systems music, in which a function "is applied repeatedly, each time taking as argument
Systems_music
Class of vascular plants
Fractals Everywhere. A self-similar structure is described by a mathematical function, applied repeatedly at different scales to create a frond pattern. The
Fern
take the form of iterated functions. Chaotic maps often occur in the study of dynamical systems. Chaotic maps and iterated functions often generate fractals
List_of_chaotic_maps
obtains the function, again, if you convolve the function with a discrete mask and then scale it back. There is a similarity to iterated function systems
Refinable_function
long as it is symmetric, we can easily think of the system as an iterated function map, a common method of representing a chaotic, dynamical system.
Chaotic_scattering
Type of fractal in mathematics
{\displaystyle c} points are iterated through the Mandelbrot function. For points which do escape within a chosen maximum number of iterations, and therefore are
Buddhabrot
Artin–Mazur zeta function, named after Michael Artin and Barry Mazur, is a function that is used for studying the iterated functions that occur in dynamical
Artin–Mazur_zeta_function
Equation for fixed point of functional composition
Ψ(x)) ≡ h1(x). In general, all of its functional iterates (its regular iteration group, see iterated function) are provided by the orbit h t ( x ) = Ψ − 1
Schröder's_equation
Number representing system complexity
by an iterated function, the topological entropy represents the exponential growth rate of the number of distinguishable orbits of the iterates. An important
Topological_entropy
Function that derives secret keys from a secret value
number which acts as cryptographic salt, and iterations refers to the number of iterations of a sub-function. The derived key is used instead of the original
Key_derivation_function
Mathematical rule
f^{(m)}(x)=x} , where f ( m ) {\displaystyle f^{(m)}} denotes the iterated function obtained by composition of m {\displaystyle m} copies of f {\displaystyle
Sharkovskii's_theorem
Trail in which only the first and last vertices are equal
even length Cycle space Cycle basis Cycle detection in a sequence of iterated function values Minimum mean weight cycle Bender & Williamson 2010, p. 164
Cycle_(graph_theory)
Each fs with s > 0 has the same Koenigs function, cf. iterated function. In fact, if h is the Koenigs function of f = f1, then h(fs(z)) satisfies Schroeder's
Koenigs_function
stack for the iterator function. */ char iterator_stack[SIGSTKSZ]; /* Flag indicating that the iterator has completed. */ volatile int iterator_finished;
Setcontext
G-function Fox H-function Hyperoperations Iterated logarithm Super-logarithms Tetration Lambert W function: Inverse of f(w) = w exp(w). Lamé function Mathieu
List of mathematical functions
List_of_mathematical_functions
Numerical approximation algorithm
the iterative process reaches sufficient accuracy already far earlier. The analysis of these methods is hard, depending on a complicated function of the
Iterative_method
Lightweight programming language
--example: print(i) end This generic for loop would iterate over the table _G using the standard iterator function pairs, until it returns nil: for key, value
Lua
Open Source fractal editor and generator
mathematical functions. Each function is a composition of an affine map, and usually some non-linear map. This set of functions is called an iterated function system
Apophysis_(software)
Form of algorithmic art
to this group. IFS (iterated function systems) Strange attractors Fractal flame L-system fractals Fractals created by the iteration of complex polynomials
Fractal_art
Metaheuristic
Iterated Local Search (ILS) is a term in applied mathematics and computer science defining a modification of local search or hill climbing methods for
Iterated_local_search
Two-dimensional fractal
for once a point has been darkened, it remains black for every other iteration; however some points remain white. The fractal dimension of the boundary
T-square_(fractal)
Type of fractal
parameters. For all iteration sequences, the diagonal a = b is always the same as for the standard one parameter logistic function. The sequence is usually
Lyapunov_fractal
Analytic function in mathematics
branch of the Lambert W-function, and B(μ) n,≥2 is an incomplete poly-Bernoulli number. The function g(x) = x(1 + ⌊x−1⌋) − 1 is iterated to find the coefficients
Riemann_zeta_function
elementary recursive function. Equivalently, these are the problems that can be solved in time bounded by an iterated exponential function with a bounded number
ELEMENTARY
Invariant measure of fractal dimension
doi:10.1214/11-STS370. S2CID 88512325. Larry Riddle, 2014, "Classic Iterated Function Systems: Koch Snowflake", Agnes Scott College e-Academy (online),
Hausdorff_dimension
Motion of particles in a fluid
to Bernoulli shifts. Abel equation Iterated function Schröder's equation Infinite compositions of analytic functions Irwin, Smooth Dynamical Systems, 1
Flow_(mathematics)
Sigmoid shape special function
{1}{2}},x^{2}\right)}.} sgn(x) is the sign function. The iterated integrals of the complementary error function are defined by i n erfc ( z ) = ∫ z ∞ i
Error_function
Polynomial function of degree two
In mathematics, a quadratic function of a single variable is a function of the form f ( x ) = a x 2 + b x + c {\displaystyle f(x)=ax^{2}+bx+c} with
Quadratic_function
Set of values for a mathematical model
c} in the complex plane which give a bounded set of numbers when a iterated function z n + 1 = f c ( z n ) = z 2 + c {\displaystyle z_{n+1}=f_{c}(z_{n})=z^{2}+c}
Parameter_space
ITERATED FUNCTION
ITERATED FUNCTION
Girl/Female
Tamil
Mokshitha | மோகà¯à®·à¯€à®¤à®¾
Liberated, Free
Mokshitha | மோகà¯à®·à¯€à®¤à®¾
Girl/Female
Muslim
Saved, Liberated
Boy/Male
Tamil
Liberated, Sage
Boy/Male
Tamil
Nirvanin | நீரà¯à®µà®¨à¯€à®¨
Liberated
Nirvanin | நீரà¯à®µà®¨à¯€à®¨
Boy/Male
Tamil
Muktananda | à®®à¯à®•à¯à®¤à®¾à®¨à®‚தா
Liberated
Muktananda | à®®à¯à®•à¯à®¤à®¾à®¨à®‚தா
Girl/Female
Hindu
Liberated, Free
Girl/Female
Hindu
Liberated, Free
Girl/Female
Tamil
Liberated, Pearl
Girl/Female
Indian
Saved, Liberated
Girl/Female
Hindu
Liberated, Pearl
Girl/Female
Hindu
Liberated, Pearl
Girl/Female
Tamil
Liberated, Pearl
Girl/Female
Tamil
Aamuktha | ஆமà¯à®•à¯à®¤à®¾Â
Liberated
Aamuktha | ஆமà¯à®•à¯à®¤à®¾Â
Girl/Female
Tamil
Mokshita | மோகà¯à®·à¯€à®¤à®¾Â
Liberated, Free
Mokshita | மோகà¯à®·à¯€à®¤à®¾Â
Boy/Male
Hindu
Liberated
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Sanskrit, Telugu
Liberated
Boy/Male
Arabic, Muslim
Liberated; Free
Girl/Female
Hindu, Indian
Liberated
Girl/Female
Indian
Liberated
Girl/Female
Muslim
Saved, Liberated
ITERATED FUNCTION
ITERATED FUNCTION
Boy/Male
Hebrew
God's grace.
Girl/Female
Hindu
Boy/Male
Tamil
Name of a prophet
Boy/Male
British, English, Gaelic, Irish
Dove; From the Woods; Diminutive of Culver
Girl/Female
Tamil
Girl/Female
Christian & English(British/American/Australian)
Helper of Mankind
Boy/Male
Spanish Hebrew English
Supplanter.
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Punjabi, Sikh
Saffron
Female
Hindi/Indian
(उषा) Hindi myth name of a demon princess, daughter of heaven, and sister of night, USHA means "dawn."
Girl/Female
English Latin Spanish American Greek
Winged.
ITERATED FUNCTION
ITERATED FUNCTION
ITERATED FUNCTION
ITERATED FUNCTION
ITERATED FUNCTION
a.
Uttered or done again; repeated.
imp. & p. p.
of Operate
p. pr. & vb. n.
of Iterate
adv.
By way of iteration.
n.
State of being literate.
v. t.
To utter or do a second time or many times; to repeat; as, to iterate advice.
imp. & p. p.
of Onerate
a.
Standardized; determined or analyzed by titration; as, titrated solutions.
a.
Reiterated; repeated.
a.
Combined, or impregnated, with nitric acid, or some of its compounds.
n.
One educated, but not having taken a university degree; especially, such a person who is prepared to take holy orders.
a.
Planet-struck; blasted.
imp. & p. p.
of Titrate
a.
Prepared with nitrate of silver.
a.
Capable of being integrated.
a.
Misapplied; treated badly.
v. t.
To keep repeating needlessly; to iterate.
n.
A literary man.
imp. & p. p.
of Iterate
a.
Capable of being iterated or repeated.