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Formally, an iterated function system is a finite set of contraction mappings on a complete metric space.

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## Some Related Sentences

Formally and function

__Formally__

**,**there

**is**

**a**clear distinction: " DFT " refers to

**a**mathematical transformation or

__function__

**,**regardless

**of**how it

**is**computed

**,**whereas " FFT " refers to

**a**specific family

**of**algorithms for computing DFTs

**.**

__Formally__

**,**if M

**is**

**a**

**set**

**,**the identity

__function__f

**on**M

**is**defined to be that

__function__with domain and codomain M which satisfies

__Formally__

**,**the discrete cosine transform

**is**

**a**linear

**,**invertible

__function__( where denotes the

**set**

**of**real numbers ), or equivalently

**an**invertible N × N square matrix

**.**

__Formally__

**,**we are given

**a**

**set**

**of**hypotheses and

**a**

**set**

**of**manifestations ; they are related by the domain knowledge

**,**represented by

**a**

__function__that takes as

**an**argument

**a**

**set**

**of**hypotheses and gives as

**a**result the corresponding

**set**

**of**manifestations

**.**

__Formally__

**,**

**a**

__function__ƒ

**is**real analytic

**on**

**an**open

**set**D in the real line if for any x < sub > 0 </ sub > in D one can write

__Formally__

**,**

**an**elliptic

__function__

**is**

**a**

__function__meromorphic

**on**for which there exist two non-zero complex numbers and with ( in other words

**,**not parallel ), such that and for all

**.**

__Formally__

**,**if

**is**

**an**open subset

**of**the complex plane

**,**

**a**point

**of**

**,**and

**is**

**a**holomorphic

__function__

**,**then

**is**called

**a**removable singularity for if there exists

**a**holomorphic

__function__which coincides with

**on**

**.**

__Formally__

**,**the problem

**of**supervised pattern recognition can be stated as follows: Given

**an**unknown

__function__( the ground truth ) that maps input instances to output labels

**,**along with training data assumed to represent accurate examples

**of**the mapping

**,**produce

**a**

__function__that approximates as closely as possible the correct mapping

**.**

__Formally__

**,**

**a**statistic s

**is**

**a**measurable

__function__

**of**X ; thus

**,**

**a**statistic s

**is**evaluated

**on**

**a**random variable X

**,**taking the value s ( X ), which

**is**itself

**a**random variable

**.**

__Formally__

**,**the discrete sine transform

**is**

**a**linear

**,**invertible

__function__F: R < sup > N </ sup > < tt >-></ tt > R < sup > N </ sup > ( where R denotes the

**set**

**of**real numbers ), or equivalently

**an**N × N square matrix

**.**

__Formally__

**,**the discrete Hartley transform

**is**

**a**linear

**,**invertible

__function__H: R < sup > n </ sup > < tt >-></ tt > R < sup > n </ sup > ( where R denotes the

**set**

**of**real numbers ).

__Formally__

**,**

**a**cardinal κ

**is**defined to be weakly compact if it

**is**uncountable and for every

__function__f: < sup > 2 </ sup > →

__Formally__

**,**

**an**analytic

__function__ƒ ( z )

**of**the real or complex variables z < sub > 1 </ sub >,…, z < sub > n </ sub >

**is**transcendental if z < sub > 1 </ sub >, …, z < sub > n </ sub >, ƒ ( z ) are algebraically independent

**,**i

**.**e., if ƒ

**is**transcendental over the field C ( z < sub > 1 </ sub >, …, z < sub > n </ sub >).

__Formally__

**,**this means that

**,**for some

__function__f

**,**the image f ( D )

**of**

**a**directed

**set**D ( i

**.**e

**.**the

**set**

**of**the images

**of**each element

**of**D )

**is**again directed and has as

**a**least upper bound the image

**of**the least upper bound

**of**D

**.**One could also say that f preserves directed suprema

**.**

__Formally__

**,**let be

**a**stochastic process and let represent the cumulative distribution

__function__

**of**the joint distribution

**of**at times

**.**

__Formally__

**,**

**an**ultrametric

**space**

**is**

**a**

**set**

**of**points with

**an**associated distance

__function__( also called

**a**

**metric**)

Formally and system

However

**,**shortly after this positive result**,**Kurt Gödel published On__Formally__Undecidable Propositions**of**Principia Mathematica and Related Systems ( 1931 ), showing that in any sufficiently strong axiomatic__system__there are true statements which cannot be proved in the__system__**.**__Formally__known as the " Six and A Half Point Pole " form

**,**this

**is**the second to last form

**of**the

__system__

**.**

__Formally__

**,**HPGCC

**is**

**a**cross-compiler ; it compiles code for the ARM-based HP calculators

**,**but runs

**on**

**a**PC rather than the target

__system__

**.**

__Formally__

**,**

**an**unlabelled state transition

__system__

**is**

**a**tuple ( S

**,**→) where S

**is**

**a**

**set**(

**of**states ) and → ⊆ S × S

**is**

**a**binary relation over S (

**of**transitions ).

__Formally__

**,**knowing the step response

**of**

**a**dynamical

__system__gives information

**on**the stability

**of**such

**a**

__system__

**,**and

**on**its ability to reach one stationary state when starting from another

**.**

__Formally__

**,**

**a**

__system__

**is**said to be observable if

**,**for any possible sequence

**of**state and control vectors

**,**the current state can be determined in

**finite**time using only the outputs ( this definition

**is**slanted towards the state

**space**representation ).

__Formally__

**,**let G be

**a**Coxeter group with reduced root

__system__R and k < sub > v </ sub >

**a**multiplicity

**function**

**on**R ( so k < sub > u </ sub > = k < sub > v </ sub > whenever the reflections σ < sub > u </ sub > and σ < sub > v </ sub > corresponding to the roots u and v are conjugate in G ).

__Formally__

**,**

**a**squeeze preserves the hyperbolic

**metric**expressed in the form ; in

**a**different coordinate

__system__

**.**

__Formally__

**,**

**an**Hamiltonian

__system__

**is**

**a**dynamical

__system__completely described by the scalar

**function**

**,**the Hamiltonian

**.**

__Formally__

**,**

**a**graph rewriting

__system__usually consists

**of**

**a**

**set**

**of**graph rewrite rules

**of**the form

**,**with being called pattern graph ( or left-hand side ) and being called replacement graph ( or right-hand side

**of**the rule ).

__Formally__

**,**Transnistria has

**a**multi-party

__system__and

**an**unicameral parliament

**,**called the Supreme Council

**.**

__Formally__

**,**

**a**redistribution

**of**all agents ’ initial endowments

**of**goods among all agents in the

__system__

**is**

**a**core allocation if no group

**of**agents can redistribute their own initial endowments among themselves so as to improve the satisfaction

**of**someone in the group without impairing that

**of**any other in the group

**.**

__Formally__known as the " Eastern Division

**of**Surrey " or " Surrey Eastern ", it elected two MPs by the bloc vote

__system__

**.**

__Formally__

**,**

**a**Markov partition

**is**used to provide

**a**

**finite**cover for the smooth

__system__; each

**set**

**of**the cover

**is**associated with

**a**single symbol

**,**and the sequences

**of**symbols result as

**a**trajectory

**of**the

__system__moves from one

**of**the covering sets to another

**.**

__Formally__

__system__networks correspond to type lattices in formal lattice theory

**,**although they are occasionally erroneously mistaken for flowcharts or directed decision trees

**.**

__Formally__known as the Federal Reserve Wire Network

**,**Fedwire

**is**

**a**Real Time Gross Settlement Funds Transfer

__system__operated by the United States Federal Reserve Banks that enables financial institutions to electronically transfer funds between its more than 9

**,**289 participants ( as

**of**March 19

**,**2009 ).

Formally and is

__Formally__organized vocational programs supported by federal funds allow high school students to gain experience in

**a**field

**of**work which

__is__likely to lead to

**a**full-time job

**on**graduation

**.**

__Formally__

**,**the

**set**

**of**all context-free languages

__is__identical to the

**set**

**of**languages accepted by pushdown automata ( PDA ).

More rigorously

**,**the divergence**of****a**vector field F at**a**point p__is__defined as the limit**of**the net flow**of**F across the smooth boundary**of****a**three dimensional region V divided by the volume**of**V as V shrinks to p**.**__Formally__**,**__Formally__

**,**the base

__is__known as Naval Support Facility Diego Garcia ( the US activity ) or Permanent Joint Operating Base ( PJOB ) Diego Garcia ( the UK's term ).

__Formally__

**,**oxidation state

__is__the hypothetical charge that

**an**atom would have if all bonds to atoms

**of**different elements were 100 % ionic

**.**

__Formally__

**,**

**an**inner product

**space**

__is__

**a**vector

**space**V over the field together with

**an**inner product

**,**i

**.**e., with

**a**map

*

__Formally__**,**when working over the reals**,**as here**,**this__is__accomplished by considering the limit as ε → 0 ; but the " infinitesimal " language generalizes directly to Lie groups over general rings**.**__Formally__

**,**

**a**profinite group

__is__

**a**Hausdorff

**,**compact

**,**and totally disconnected topological group: that

__is__

**,**

**a**topological group that

__is__also

**a**Stone

**space**

**.**

__Formally__

**,**this sharing

**of**dynamics

__is__referred to as universality

**,**and systems with precisely the same critical exponents are said to belong to the same universality class

**.**

__Formally__

**,**

**a**frame

__is__defined to be

**a**lattice L in which

**finite**meets distribute over arbitrary joins

**,**i

**.**e

**.**every ( even infinite ) subset

0.115 seconds.