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Formally, a deterministic algorithm computes a mathematical function ; a function has a unique value for any given input, and the algorithm is a process that produces this particular value as output.

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## Some Related Sentences

Formally and deterministic

__Formally__

**,**

**a**

__deterministic__Büchi automaton

**is**

**a**tuple A = ( Q

**,**Σ

**,**δ

**,**q < sub > 0 </ sub >, F )

**that**consists of

**the**following components:

__Formally__

**,**

**a**

__deterministic__finite automaton may be defined by

**the**tuple where

**is**

**the**set of states of

**the**automaton

**,**

**is**

**the**set of

**input**symbols

**,**

**is**

**the**transition

**function**

**that**takes

**a**state

**and**an

**input**symbol to

**a**new state

**,**

**is**

**the**initial state of

**the**automaton

**,**

**and**

**is**

**the**set of accepting or final states of

**the**automaton

**.**

Formally and algorithm

Formally and mathematical

__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__

**,**dependence refers to

**any**situation in which random variables do not satisfy

**a**

__mathematical__condition of probabilistic independence

**.**

Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I (" On

__Formally__Undecidable Propositions of Principia Mathematica**and**Related Systems I ")**is****a**paper in__mathematical__logic by Kurt Gödel**.**__Formally__

**,**consider an economic model with different

__mathematical__weightings placed on

**the**utilities of each self

**.**

Formally and function

__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 ;

*

__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__

**,**these failed when they were rejected by

**the**Church of England's General Synod in 1972

__;__conversations

**and**co-operation continued

**,**however

**,**leading in 2003 to

**the**signing of

**a**covenant between

**the**two churches

**.**

__Formally__

**,**

**the**case where only

**a**subset of parameters

**is**defined

**is**still

**a**composite hypothesis

__;__nonetheless

**,**

**the**term point hypothesis

**is**often applied in such cases

**,**particularly where

**the**hypothesis test can be structured in such

**a**way

**that**

**the**distribution of

**the**test statistic (

**the**distribution under

**the**null hypothesis ) does not depend on

**the**parameters whose values have not been specified under

**the**point null hypothesis

**.**

__;__Electronics:

__Formally__

**a**fixture in radio shacks

**,**now used

**for**assembly

**and**repair of all sorts of electronic equipment including communication

**,**computer

**,**

**and**home entertainment items

**.**

__Formally__

**,**Aff ( V )

**is**naturally isomorphic to

**a**subgroup of

**,**with V embedded

**as**

**the**affine plane

**,**namely

**the**stabilizer of

**this**affine plane

__;__

**the**above matrix formulation

**is**

**the**( transpose of )

**the**realization of

**this**

**,**with

**the**( n × n

**and**1 × 1 ) blocks corresponding to

**the**direct sum decomposition

**.**

__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__

**,**Korea's governmental FTC ( Fair Trade Commission: 공정거래위원회

__;__公正去來委員會 ) announces

**a**limited Chaebol list every year

**as**size of industrial assets ( not including financial companies ).

__Formally__

**the**euro

**is**

**the**only legal tender in Büsingen

__;__however

**,**

**the**Swiss franc

**is**far more popular in everyday use

**as**shopping

**is**generally done in nearby Schaffhausen instead of across

**the**border ( s ) in Germany

**.**

__Formally__

**,**

**the**algorithm's performance will be

**a**random variable determined by

**the**random bits

__;__thus either

**the**running time

**,**or

**the**

**output**( or both ) are random variables

**.**

__Formally__

**,**Chrétien

**and**Martin have remained publicly respectful of each other

__;__while much of

**the**verbal sparring was between their supporters

**.**

__Formally__

**,**

**a**squeeze preserves

**the**hyperbolic metric expressed in

**the**form

__;__in

**a**different coordinate system

**.**

Bābak Khorram-Din (

__Formally__known**as**" Pāpak " meaning " Young Father ") (, alternative spelling: Bābak Xorramdin__;__795**,**according to some other sources 798 — January 838 ) was one of**the**main PersianArmenian historian Vardan Areweltsʻi**,**ca**.**__Formally__described

**,**it alternates slow sections in

**a**modal F with faster sections

**,**" Neue Kraft fühlend " ( with renewed strength ), in D

**.**The slow sections each have two elements

**,**( 1 )

**a**passage reminiscent of

**the**opening of

**the**first movement in which

**the**instruments overlap each other with

**a**brief motive

__;__( 2 )

**a**chorale

**,**

**the**actual song

**.**

__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__

**,**citizen science

**has**been defined

**as**"

**the**systematic collection

**and**analysis of data

__;__development of technology

__;__testing of natural phenomena

__;__

**and**

**the**dissemination of these activities by researchers on

**a**primarily avocational basis ".

__Formally__

**,**at least

**,**

**the**European Commission

**has**primarily

**a**monitoring role

__;__in practice

**,**however

**,**there

**is**considerable scope

**for**it to help set

**the**policy agenda

**and**persuade reluctant Member States to implement agreed policies

**.**

__Formally__

**,**let P be

**a**poset ( partially ordered set ),

**and**let F be

**a**filter on P

__;__

**that**

**is**

**,**F

**is**

**a**subset of P such

**that**:

__Formally__

**,**

**a**biased graph Ω

**is**

**a**pair ( G

**,**B ) where B

**is**

**a**linear class of circles

__;__

**this**by definition

**is**

**a**class of circles

**that**satisfies

**the**theta-graph property mentioned above

**.**

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